The Mizar article:

Hahn-Banach Theorem

by
Bogdan Nowak, and
Andrzej Trybulec

Received July 8, 1993

Copyright (c) 1993 Association of Mizar Users

MML identifier: HAHNBAN
[ MML identifier index ]


environ

 vocabulary FUNCT_1, RELAT_1, BOOLE, PARTFUN1, FRAENKEL, TARSKI, SUPINF_1,
      ARYTM_3, ORDINAL2, RLVECT_1, RLSUB_1, RLSUB_2, MCART_1, RLVECT_3,
      GRCAT_1, UNIALG_1, ABSVALUE, FUNCOP_1, ARYTM_1, SEQ_2, NORMSP_1, HAHNBAN,
      ARYTM;
 notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, XREAL_0, STRUCT_0,
      REAL_1, ABSVALUE, RELAT_1, FUNCT_1, FUNCT_2, RLVECT_1, RLSUB_1, RLSUB_2,
      NORMSP_1, PARTFUN1, FUNCOP_1, RFUNCT_3, RLVECT_3, FRAENKEL, DOMAIN_1,
      SUPINF_1;
 constructors REAL_1, ABSVALUE, RLSUB_2, NORMSP_1, RFUNCT_3, RLVECT_2,
      RLVECT_3, DOMAIN_1, MEASURE5, MEMBERED, XBOOLE_0;
 clusters SUBSET_1, FRAENKEL, RLVECT_1, RLSUB_1, SUPINF_1, RELSET_1, STRUCT_0,
      XREAL_0, PARTFUN1, MEMBERED, ZFMISC_1, XBOOLE_0;
 requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
 definitions TARSKI, RLSUB_1, FRAENKEL, SUPINF_1, RLSUB_2, RLVECT_1, ORDINAL1,
      XBOOLE_0;
 theorems PARTFUN1, RLSUB_1, FUNCT_2, ZFMISC_1, FUNCOP_1, REAL_1, AXIOMS,
      TREES_2, TARSKI, FUNCT_1, FRAENKEL, GRFUNC_1, RLVECT_1, RLSUB_2,
      RLVECT_4, SUPINF_1, MCART_1, RLVECT_3, ABSVALUE, NORMSP_1, RELAT_1,
      RELSET_1, XREAL_0, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1;
 schemes ORDERS_2, DOMAIN_1, FUNCT_2;

begin

canceled;

theorem Th2:
 for X being set, f,g being Function st X c= dom f & f c= g
  holds f|X = g|X
 proof let X be set, f,g be Function such that
A1:  X c= dom f;
  assume f c= g;
  hence f|X = g|(dom f)|X by GRFUNC_1:64
    .= g|((dom f) /\ X) by RELAT_1:100
    .= g|X by A1,XBOOLE_1:28;
 end;

theorem Th3:
 for A being non empty set, b being set st A <> {b}
  ex a being Element of A st a <> b
 proof let A be non empty set, b be set such that
A1: A <> {b};
  assume
A2: for a being Element of A holds a = b;
     now let a be set;
    thus a in A implies a = b by A2;
    assume
A3:   a = b;
     consider a0 being Element of A;
       a0 = b by A2;
    hence a in A by A3;
   end;
  hence contradiction by A1,TARSKI:def 1;
 end;

theorem Th4:
 for X,Y being set, B being non empty Subset of PFuncs(X,Y)
  holds B is non empty functional set
 proof let X,Y be set, B be non empty Subset of PFuncs(X,Y);
    B is functional
   proof let x be set;
    assume x in B;
     hence x is Function by PARTFUN1:120;
   end;
  hence thesis;
 end;

theorem Th5:
 for B being non empty functional set, f being Function st f = union B holds
  dom f = union { dom g where g is Element of B: not contradiction } &
  rng f = union { rng g where g is Element of B: not contradiction }
 proof let B be non empty functional set, f be Function such that
A1: f = union B;
   set X = { dom g where g is Element of B: not contradiction };
     now let x be set;
    hereby assume x in dom f;
     then [x,f.x] in f by FUNCT_1:8;
     then consider g being set such that
A2:    [x,f.x] in g & g in B by A1,TARSKI:def 4;
      reconsider g as Function by A2,FRAENKEL:def 1;
     take Z = dom g;
     thus x in Z & Z in X by A2,FUNCT_1:8;
    end;
    given Z being set such that
A3:   x in Z & Z in X;
     consider g being Element of B such that
A4:   Z = dom g by A3;
       [x,g.x] in g by A3,A4,FUNCT_1:8;
     then [x,g.x] in f by A1,TARSKI:def 4;
    hence x in dom f by FUNCT_1:8;
   end;
  hence dom f = union X by TARSKI:def 4;
   set X = { rng g where g is Element of B: not contradiction };
     now let y be set;
    hereby assume y in rng f;
     then consider x being set such that
A5:    x in dom f & y = f.x by FUNCT_1:def 5;
       [x,y] in f by A5,FUNCT_1:8;
     then consider g being set such that
A6:    [x,y] in g & g in B by A1,TARSKI:def 4;
      reconsider g as Function by A6,FRAENKEL:def 1;
     take Z = rng g;
        x in dom g & y = g.x by A6,FUNCT_1:8;
     hence y in Z & Z in X by A6,FUNCT_1:def 5;
    end;
    given Z being set such that
A7:   y in Z & Z in X;
     consider g being Element of B such that
A8:   Z = rng g by A7;
     consider x being set such that
A9:    x in dom g & y = g.x by A7,A8,FUNCT_1:def 5;
       [x,y] in g by A9,FUNCT_1:8;
     then [x,y] in f by A1,TARSKI:def 4;
    hence y in rng f by RELAT_1:20;
   end;
  hence rng f = union X by TARSKI:def 4;
 end;

theorem
   for A being non empty Subset of ExtREAL st
  for r being R_eal st r in A holds r <=' -infty
 holds A = {-infty}
 proof let A be non empty Subset of ExtREAL such that
A1: for r being R_eal st r in A holds r <=' -infty;
  assume A <> {-infty};
   then consider a being Element of A such that
A2:  a <> -infty by Th3;
     a <=' -infty by A1;
  hence contradiction by A2,SUPINF_1:23;
 end;

theorem
   for A being non empty Subset of ExtREAL st
  for r being R_eal st r in A holds +infty <=' r
 holds A = {+infty}
 proof let A be non empty Subset of ExtREAL such that
A1: for r being R_eal st r in A holds +infty <=' r;
  assume A <> {+infty};
   then consider a being Element of A such that
A2:  a <> +infty by Th3;
     +infty <=' a by A1;
  hence contradiction by A2,SUPINF_1:24;
 end;

theorem Th8:
 for A being non empty Subset of ExtREAL, r being R_eal
  st r <' sup A
  ex s being R_eal st s in A & r <' s
 proof let A be non empty Subset of ExtREAL, r be R_eal;
  assume A1: r <' sup A;
  assume
A2: for s being R_eal st s in A holds not r <' s;
     r is majorant of A
    proof let x be R_eal;
     assume x in A;
     hence x <=' r by A2;
    end;
  hence contradiction by A1,SUPINF_1:def 16;
 end;

theorem Th9:
 for A being non empty Subset of ExtREAL, r being R_eal
  st inf A <' r
  ex s being R_eal st s in A & s <' r
 proof let A be non empty Subset of ExtREAL, r be R_eal;
  assume A1: inf A <' r;
  assume
A2: for s being R_eal st s in A holds not s <' r;
     r is minorant of A
    proof let x be R_eal;
     assume x in A;
     hence r <=' x by A2;
    end;
  hence contradiction by A1,SUPINF_1:def 17;
 end;

theorem Th10:
 for A,B being non empty Subset of ExtREAL st
  for r,s being R_eal st r in A & s in B holds r <=' s
  holds sup A <=' inf B
 proof let A,B be non empty Subset of ExtREAL;
  assume that
A1: for r,s being R_eal st r in A & s in B holds r <=' s;
  assume not sup A <=' inf B;
   then consider s1 being R_eal such that
A2:  s1 in A & inf B <' s1 by Th8;
   consider s2 being R_eal such that
A3:  s2 in B & s2 <' s1 by A2,Th9;
  thus contradiction by A1,A2,A3;
 end;

canceled;

theorem Th12:
   for x,y being R_eal, p,q being real number st
   x = p & y = q holds (p <= q iff x <=' y)
proof
   let x,y be R_eal;
   let p,q be real number;
   assume
A1:x = p & y = q;
   reconsider p, q as Real by XREAL_0:def 1;
A2:x = p & y = q by A1;
     x <=' y implies p <= q
   proof
      assume x <=' y;
      then consider p1,q1 being Real such that
   A3:p1 = x & q1 = y & p1 <= q1 by A2,SUPINF_1:16;
      thus thesis by A1,A3;
   end;
   hence thesis by A2,SUPINF_1:16;
end;

begin :: Chains of Functions

definition let A be non empty set;
 cluster c=-linear non empty Subset of A;
 existence
  proof consider a being Element of A;
    reconsider B = { a } as non empty Subset of A;
   take B;
      B is c=-linear
     proof let x,y be set;
      assume x in B & y in B;
       then x = a & y = a by TARSKI:def 1;
      hence thesis;
     end;
   hence thesis;
  end;
end;

theorem Th13:
 for X,Y being set
  for B being c=-linear Subset of PFuncs(X,Y)
   holds union B in PFuncs(X,Y)
 proof let X,Y be set; let B be c=-linear Subset of PFuncs(X,Y);
   for x be set st x in B holds x is Function by PARTFUN1:120;
   then reconsider f = union B as Function by TREES_2:36;
   per cases;
   suppose B <> {};
   then reconsider D = B as non empty functional set by Th4;
A1: now let x be set;
    assume x in { dom g where g is Element of D: not contradiction };
     then consider g being Element of D such that
A2:    x = dom g;
       g in PFuncs(X,Y) by TARSKI:def 3;
     then ex f being Function st g = f & dom f c= X & rng f c= Y
      by PARTFUN1:def 5;
    hence x c= X by A2;
   end;
A3: now let x be set;
    assume x in { rng g where g is Element of D: not contradiction };
     then consider g being Element of D such that
A4:    x = rng g;
       g in PFuncs(X,Y) by TARSKI:def 3;
     then ex f being Function st g = f & dom f c= X & rng f c= Y
      by PARTFUN1:def 5;
    hence x c= Y by A4;
   end;
     dom f = union { dom g where g is Element of D: not contradiction } &
   rng f = union { rng g where g is Element of D: not contradiction } by Th5;
   then dom f c= X & rng f c= Y by A1,A3,ZFMISC_1:94;
  hence union B in PFuncs(X,Y) by PARTFUN1:def 5;
  suppose A5: B = {};
    {} is PartFunc of X, Y by PARTFUN1:56;
  hence thesis by A5,PARTFUN1:119,ZFMISC_1:2;
 end;

begin :: Some Facts on Real Vector Spaces

reserve V for RealLinearSpace;

theorem Th14:
 for W1,W2 being Subspace of V
  holds the carrier of W1 c= the carrier of W1 + W2
 proof let W1,W2 be Subspace of V;
  let x be set;
  assume
A1: x in the carrier of W1;
     the carrier of W1 c= the carrier of V by RLSUB_1:def 2;
   then reconsider w = x as VECTOR of V by A1;
A2: w + 0.V = w by RLVECT_1:10;
     x in W1 & 0.V in W2 by A1,RLSUB_1:25,RLVECT_1:def 1;
   then x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by A2;
  hence x in the carrier of W1 + W2 by RLSUB_2:def 1;
 end;

theorem Th15:
 for W1,W2 being Subspace of V
  st V is_the_direct_sum_of W1,W2
 for v,v1,v2 being VECTOR of V st v1 in W1 & v2 in W2 & v = v1 + v2
  holds v |-- (W1,W2) = [v1,v2]
 proof let W1,W2 be Subspace of V;
  assume
A1: V is_the_direct_sum_of W1,W2;
  let v,v1,v2 be VECTOR of V;
     [v1,v2]`1 = v1 & [v1,v2]`2 = v2 by MCART_1:7;
  hence thesis by A1,RLSUB_2:def 6;
 end;

theorem Th16:
 for W1,W2 being Subspace of V
  st V is_the_direct_sum_of W1,W2
 for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2]
  holds v = v1 + v2
 proof let W1,W2 be Subspace of V such that
A1: V is_the_direct_sum_of W1,W2;
  let v,v1,v2 be VECTOR of V;
  assume v |-- (W1,W2) = [v1,v2];
   then (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2 by MCART_1:7;
  hence v = v1 + v2 by A1,RLSUB_2:def 6;
 end;

theorem Th17:
 for W1,W2 being Subspace of V
  st V is_the_direct_sum_of W1,W2
 for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2]
  holds v1 in W1 & v2 in W2
 proof let W1,W2 be Subspace of V such that
A1: V is_the_direct_sum_of W1,W2;
  let v,v1,v2 be VECTOR of V;
  assume v |-- (W1,W2) = [v1,v2];
   then (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2 by MCART_1:7;
  hence thesis by A1,RLSUB_2:def 6;
 end;

theorem Th18:
 for W1,W2 being Subspace of V
  st V is_the_direct_sum_of W1,W2
 for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2]
  holds v |-- (W2,W1) = [v2,v1]
 proof let W1,W2 be Subspace of V;
  assume
A1: V is_the_direct_sum_of W1,W2;
then A2: V is_the_direct_sum_of W2,W1 by RLSUB_2:46;
  let v,v1,v2 be VECTOR of V;
  assume v |-- (W1,W2) = [v1,v2];
    then A3:   (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2 by MCART_1:7;
    then A4:   v1 in W1 & v2 in W2 by A1,RLSUB_2:def 6;
      v = v2 + v1 by A1,A3,RLSUB_2:def 6;
  hence v |-- (W2,W1) = [v2,v1] by A2,A4,Th15;
 end;

theorem Th19:
 for W1,W2 being Subspace of V
  st V is_the_direct_sum_of W1,W2
 for v being VECTOR of V st v in W1 holds v |-- (W1,W2) = [v,0.V]
  proof let W1,W2 be Subspace of V such that
A1: V is_the_direct_sum_of W1,W2;
   let v be VECTOR of V such that
A2: v in W1;
A3:  0.V in W2 by RLSUB_1:25;
      v + 0.V = v by RLVECT_1:10;
   hence v |-- (W1,W2) = [v,0.V] by A1,A2,A3,Th15;
  end;

theorem Th20:
 for W1,W2 being Subspace of V
  st V is_the_direct_sum_of W1,W2
 for v being VECTOR of V st v in W2 holds v |-- (W1,W2) = [0.V,v]
 proof let W1,W2 be Subspace of V;
  assume V is_the_direct_sum_of W1,W2;
   then A1:  V is_the_direct_sum_of W2,W1 by RLSUB_2:46;
  let v be VECTOR of V;
   assume v in W2;
    then v |-- (W2,W1) = [v,0.V] by A1,Th19;
   hence v |-- (W1,W2) = [0.V,v] by A1,Th18;
 end;

theorem Th21:
 for V1 being Subspace of V, W1 being Subspace of V1,
  v being VECTOR of V st v in W1 holds v is VECTOR of V1
 proof let V1 be Subspace of V, W1 be Subspace of V1,
  v be VECTOR of V such that
A1: v in W1;
A2: the carrier of W1 c= the carrier of V1 by RLSUB_1:def 2;
     v in the carrier of W1 by A1,RLVECT_1:def 1;
  hence v is VECTOR of V1 by A2;
 end;

theorem Th22:
 for V1,V2,W being Subspace of V, W1,W2 being Subspace of W
  st W1 = V1 & W2 = V2
 holds W1 + W2 = V1 + V2
 proof let V1,V2,W be Subspace of V, W1,W2 be Subspace of W such that
A1: W1 = V1 & W2 = V2;
   reconsider W3 = W1 + W2 as Subspace of V by RLSUB_1:35;
     now let v be VECTOR of V;
    hereby assume
A2:     v in W3;
      then reconsider w = v as VECTOR of W by Th21;
      consider w1,w2 being VECTOR of W such that
A3:    w1 in W1 & w2 in W2 and
A4:    w = w1 + w2 by A2,RLSUB_2:5;
      reconsider v1 = w1, v2 = w2 as VECTOR of V by RLSUB_1:18;
        v = v1 + v2 by A4,RLSUB_1:21;
     hence v in V1 + V2 by A1,A3,RLSUB_2:5;
    end;
A5:  the carrier of W1 c= the carrier of W &
    the carrier of W2 c= the carrier of W by RLSUB_1:def 2;
    assume v in V1 + V2;
     then consider v1,v2 being VECTOR of V such that
A6:    v1 in V1 & v2 in V2 and
A7:    v = v1 + v2 by RLSUB_2:5;
        v1 in the carrier of W1 & v2 in the carrier of W2
        by A1,A6,RLVECT_1:def 1;
      then reconsider w1 = v1, w2 = v2 as VECTOR of W by A5;
        v = w1 + w2 by A7,RLSUB_1:21;
    hence v in W3 by A1,A6,RLSUB_2:5;
   end;
  hence W1 + W2 = V1 + V2 by RLSUB_1:39;
 end;

theorem Th23:
 for W being Subspace of V
 for v being VECTOR of V, w being VECTOR of W st v = w
 holds Lin{w} = Lin{v}
 proof let W be Subspace of V;
  let v be VECTOR of V, w be VECTOR of W such that
A1: v = w;
   reconsider W1 = Lin{w} as Subspace of V by RLSUB_1:35;
     now let u be VECTOR of V;
    hereby assume u in W1;
      then consider a being Real such that
A2:     u = a * w by RLVECT_4:11;
        u = a * v by A1,A2,RLSUB_1:22;
     hence u in Lin{v} by RLVECT_4:11;
    end;
    assume u in Lin{v};
     then consider a being Real such that
A3:     u = a * v by RLVECT_4:11;
       u = a * w by A1,A3,RLSUB_1:22;
    hence u in W1 by RLVECT_4:11;
   end;
  hence Lin{w} = Lin{v} by RLSUB_1:39;
 end;

theorem Th24:
 for v being VECTOR of V, X being Subspace of V st not v in X
 for y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v}
  st v = y & W = X
 holds X + Lin{v} is_the_direct_sum_of W,Lin{y}
 proof let v be VECTOR of V, X be Subspace of V such that
A1: not v in X;
  let y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; assume
A2: v = y & W = X;
   then Lin{v} = Lin{y} by Th23;
  hence the RLSStruct of X + Lin{v} = W + Lin{y} by A2,Th22;
  assume W /\ Lin{y} <> (0).(X + Lin{v});
   then consider z being VECTOR of X + Lin{v} such that
A3:  not(z in W /\ Lin{y} iff z in (0).(X + Lin{v})) by RLSUB_1:39;
  per cases by A3;
  suppose that
A4: z in W /\ Lin{y} and
A5: not z in (0).(X + Lin{v});
A6: z in W by A4,RLSUB_2:7;
     z in Lin{y} by A4,RLSUB_2:7;
   then consider a being Real such that
A7: z = a * y by RLVECT_4:11;
     now per cases;
    suppose a = 0;
      then z = 0.(X + Lin{v}) by A7,RLVECT_1:23;
     hence contradiction by A5,RLSUB_1:25;
    suppose
A8:   a <> 0;
        y = 1*y by RLVECT_1:def 9 .= a"*a*y by A8,XCMPLX_0:def 7
       .= a"*(a*y) by RLVECT_1:def 9;
     hence contradiction by A1,A2,A6,A7,RLSUB_1:29;
   end;
  hence contradiction;
  suppose that
A9: not z in W /\ Lin{y} and
A10: z in (0).(X + Lin{v});
     z = 0.(X + Lin{v}) by A10,RLVECT_3:36;
  hence contradiction by A9,RLSUB_1:25;
 end;

theorem Th25:
 for v being VECTOR of V, X being Subspace of V,
   y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v}
   st v = y & X = W & not v in X
 holds y |-- (W,Lin{y}) = [0.W,y]
 proof
  let v be VECTOR of V, X be Subspace of V,
   y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v};
A1: y in Lin{y} by RLVECT_4:12;
  assume
   v = y & X = W & not v in X;
    then X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th24;
   then y |-- (W,Lin{y}) = [0.(X + Lin{v}),y] by A1,Th20;
  hence y |-- (W,Lin{y}) = [0.W,y] by RLSUB_1:19;
 end;

theorem Th26:
 for v being VECTOR of V, X being Subspace of V,
   y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v}
   st v = y & X = W & not v in X
 for w being VECTOR of X + Lin{v} st w in X
  holds w |-- (W,Lin{y}) = [w,0.V]
 proof
  let v be VECTOR of V, X be Subspace of V,
   y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v} such that
A1: v = y & X = W & not v in X;
  let w be VECTOR of X + Lin{v} such that
A2: w in X;
      X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,Th24;
   then w |-- (W,Lin{y}) = [w,0.(X + Lin{v})] by A1,A2,Th19;
  hence w |-- (W,Lin{y}) = [w,0.V] by RLSUB_1:19;
 end;

theorem Th27:
 for v being VECTOR of V, W1,W2 being Subspace of V
 ex v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2]
 proof let v be VECTOR of V, W1,W2 be Subspace of V;
   take (v |-- (W1,W2))`1,(v |-- (W1,W2))`2;
  thus v |-- (W1,W2) = [(v |-- (W1,W2))`1,v |-- (W1,W2)`2] by MCART_1:23;
 end;

theorem Th28:
 for v being VECTOR of V, X being Subspace of V,
   y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v}
   st v = y & X = W & not v in X
 for w being VECTOR of X + Lin{v}
  ex x being VECTOR of X, r being Real st w |-- (W,Lin{y}) = [x,r*v]
 proof let v be VECTOR of V, X be Subspace of V,
   y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v} such that
A1: v = y & X = W & not v in X;
  let w be VECTOR of X + Lin{v};
A2:  X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,Th24;
   consider v1,v2 being VECTOR of X + Lin{v} such that
A3:  w |-- (W,Lin{y}) = [v1,v2] by Th27;
     v1 in W by A2,A3,Th17;
   then reconsider x = v1 as VECTOR of X by A1,RLVECT_1:def 1;
     v2 in Lin{y} by A2,A3,Th17;
   then consider r being Real such that
A4:  v2 = r * y by RLVECT_4:11;
  take x,r;
  thus w |-- (W,Lin{y}) = [x,r*v] by A1,A3,A4,RLSUB_1:22;
 end;

theorem Th29:
 for v being VECTOR of V, X being Subspace of V,
   y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v}
   st v = y & X = W & not v in X
 for w1,w2 being VECTOR of X + Lin{v}, x1,x2 being VECTOR of X,
     r1,r2 being Real
  st w1 |-- (W,Lin{y}) = [x1,r1*v] & w2 |-- (W,Lin{y}) = [x2,r2*v]
  holds w1 + w2 |-- (W,Lin{y}) = [x1 + x2, (r1+r2)*v]
 proof let v be VECTOR of V, X be Subspace of V,
   y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; assume
A1: v = y & X = W & not v in X;
then A2: X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th24;
  let w1,w2 be VECTOR of X + Lin{v}, x1,x2 be VECTOR of X,
     r1,r2 be Real such that
A3: w1 |-- (W,Lin{y}) = [x1,r1*v] & w2 |-- (W,Lin{y}) = [x2,r2*v];
   reconsider z1 = x1, z2 = x2 as VECTOR of V by RLSUB_1:18;
   reconsider y1 = x1, y2 = x2 as VECTOR of X + Lin{v} by A1,RLSUB_1:18;
A4:  r1*v = r1*y & r2*v = r2*y by A1,RLSUB_1:22;
    then y1 in W & y2 in W by A2,A3,Th17;
then A5: y1 + y2 in W by RLSUB_1:28;
       (r1+r2)*v = (r1+r2)*y by A1,RLSUB_1:22;
then A6: (r1+r2)*v in Lin{y} by RLVECT_4:11;
       w1 = y1 + r1*y & w2 = y2 + r2*y by A2,A3,A4,Th16;
     then A7: w1 + w2 = y1 + r1*y + y2 + r2*y by RLVECT_1:def 6
        .= y1 + y2 + r1*y + r2*y by RLVECT_1:def 6
        .= y1 + y2 + (r1*y + r2*y) by RLVECT_1:def 6
        .= y1 + y2 + (r1+r2)*y by RLVECT_1:def 9;
A8: (r1+r2)*y = (r1+r2)*v by A1,RLSUB_1:22;
     y1 + y2 = z1 + z2 by RLSUB_1:21 .= x1 + x2 by RLSUB_1:21;
  hence w1 + w2 |-- (W,Lin{y}) = [x1 + x2, (r1+r2)*v] by A2,A5,A6,A7,A8,Th15;
 end;

theorem Th30:
 for v being VECTOR of V, X being Subspace of V,
   y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v}
   st v = y & X = W & not v in X
 for w being VECTOR of X + Lin{v}, x being VECTOR of X, t,r being Real
  st w |-- (W,Lin{y}) = [x,r*v]
  holds t*w |-- (W,Lin{y}) = [t*x, t*r*v]
 proof let v be VECTOR of V, X be Subspace of V,
   y be VECTOR of X + Lin{v}, W be Subspace of X + Lin{v}; assume
A1: v = y & X = W & not v in X;
then A2: X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th24;
  let w be VECTOR of X + Lin{v}, x be VECTOR of X, t,r be Real such that
A3: w |-- (W,Lin{y}) = [x,r*v];
   reconsider z = x as VECTOR of X + Lin{v} by A1,RLSUB_1:18;
   reconsider u = x as VECTOR of V by RLSUB_1:18;
A4: t*z = t*u by RLSUB_1:22 .= t*x by RLSUB_1:22;
A5: t*r*y = t*r*v by A1,RLSUB_1:22;
    A6: t*z in W by A1,A4,RLVECT_1:def 1;
A7: t*r*y in Lin{y} by RLVECT_4:11;
     r*y = r*v by A1,RLSUB_1:22;
   then t*w = t*(z + r*y) by A2,A3,Th16
     .= t*z + t*(r*y) by RLVECT_1:def 9
     .= t*z + t*r*y by RLVECT_1:def 9;
  hence t*w |-- (W,Lin{y}) = [t*x, t*r*v] by A2,A4,A5,A6,A7,Th15;
 end;

begin :: Functionals in Real Linear Space

definition
  let V be RLSStruct;
 canceled 2;

  mode Functional of V is Function of the carrier of V,REAL;
end;

definition
let V; let IT be Functional of V;
attr IT is subadditive means
:Def3:  for x,y being VECTOR of V holds IT.(x+y) <= IT.x+IT.y;
attr IT is additive means
:Def4:  for x,y being VECTOR of V holds IT.(x+y) = IT.x+IT.y;
attr IT is homogeneous means
:Def5:   for x being VECTOR of V, r being Real
    holds IT.(r*x) = r*IT.x;
attr IT is positively_homogeneous means
:Def6:   for x being VECTOR of V, r being Real st r > 0
    holds IT.(r*x) = r*IT.x;
attr IT is semi-homogeneous means
:Def7:   for x being VECTOR of V, r being Real st r >= 0
    holds IT.(r*x) = r*IT.x;
attr IT is absolutely_homogeneous means
:Def8:   for x being VECTOR of V, r being Real
    holds IT.(r*x) = abs(r)*IT.x;
attr IT is 0-preserving means
:Def9: IT.(0.V) = 0;
end;

definition
let V;
cluster additive -> subadditive Functional of V;
  coherence
   proof let f be Functional of V;
    assume
A1:     f is additive;
    let x,y be VECTOR of V;
     thus f.(x+y) <= f.x+f.y by A1,Def4;
   end;
cluster homogeneous -> positively_homogeneous Functional of V;
  coherence
   proof let f be Functional of V;
    assume A2: f is homogeneous;
    let x be VECTOR of V, r be Real;
    thus thesis by A2,Def5;
   end;
cluster semi-homogeneous -> positively_homogeneous Functional of V;
  coherence
   proof let f be Functional of V;
    assume A3: f is semi-homogeneous;
    let x be VECTOR of V, r be Real;
    assume r > 0;
    hence thesis by A3,Def7;
   end;
cluster semi-homogeneous -> 0-preserving Functional of V;
  coherence
   proof let f be Functional of V;
    assume A4: f is semi-homogeneous;
    thus f.(0.V) = f.(0*0.V) by RLVECT_1:23
         .= 0*f.(0.V) by A4,Def7 .= 0;
   end;
cluster absolutely_homogeneous -> semi-homogeneous Functional of V;
  coherence
   proof let f be Functional of V;
    assume A5: f is absolutely_homogeneous;
    let x be VECTOR of V, r be Real;
    assume r >= 0;
     then abs r = r by ABSVALUE:def 1;
    hence thesis by A5,Def8;
   end;
 cluster
  0-preserving positively_homogeneous -> semi-homogeneous Functional of V;
 coherence
  proof let f be Functional of V such that
A6: f is 0-preserving and
A7: f is positively_homogeneous;
   let x be VECTOR of V, r be Real such that
A8: r >= 0;
   per cases by A8;
   suppose
A9:  r = 0;
   hence f.(r*x) = f.(0.V) by RLVECT_1:23
     .= r*f.x by A6,A9,Def9;
   suppose r > 0;
   hence f.(r*x) = r*f.x by A7,Def6;
  end;
end;

definition
let V;
cluster additive absolutely_homogeneous homogeneous Functional of V;
  existence
  proof
    reconsider f = (the carrier of V) --> 0 as Functional of V by FUNCOP_1:58;
   take f;
   hereby let x,y be VECTOR of V;
    thus f.(x+y) = 0+0 by FUNCOP_1:13
         .= f.x+0 by FUNCOP_1:13
         .= f.x+f.y by FUNCOP_1:13;
   end;
   hereby let x be VECTOR of V, r be Real;
    thus f.(r*x) = abs(r)*0 by FUNCOP_1:13
       .= abs(r)*f.x by FUNCOP_1:13;
   end;
   let x be VECTOR of V, r be Real;
   thus f.(r*x) = r*0 by FUNCOP_1:13
      .= r*f.x by FUNCOP_1:13;
  end;
end;

definition
let V;
mode Banach-Functional of V is
subadditive positively_homogeneous Functional of V;
mode linear-Functional of V is
additive homogeneous Functional of V;
end;

theorem Th31:
 for L being homogeneous Functional of V, v being VECTOR of V
  holds L.(-v) = - L.v
proof let L be homogeneous Functional of V, v be VECTOR of V;
 thus L.(-v) = L.((-1)*v) by RLVECT_1:29
     .= (-1)*L.v by Def5 .= - L.v by XCMPLX_1:180;
end;

theorem Th32:
 for L being linear-Functional of V, v1,v2 being VECTOR of V
  holds L.(v1 - v2) = L.v1 - L.v2
proof let L be linear-Functional of V, v1,v2 be VECTOR of V;
 thus L.(v1 - v2) = L.(v1 + -v2) by RLVECT_1:def 11
   .= L.(v1) + L.(-v2) by Def4
   .= L.v1 + - L.v2 by Th31
   .= L.v1 - L.v2 by XCMPLX_0:def 8;
end;

theorem Th33:
 for L being additive Functional of V holds L.(0.V) = 0
proof let L be additive Functional of V;
    L.(0.V) + 0 = L.(0.V + 0.V) by RLVECT_1:10
   .= L.(0.V) + L.(0.V) by Def4;
 hence L.(0.V) = 0 by XCMPLX_1:2;
end;

theorem Th34:
 for X being Subspace of V, fi being linear-Functional of X,
     v being VECTOR of V, y being VECTOR of X + Lin{v} st v = y & not v in X
     for r being Real
      ex psi being linear-Functional of X + Lin{v} st
       psi|the carrier of X=fi & psi.y = r
proof let X be Subspace of V,
  fi be linear-Functional of X, v be VECTOR of V, y be VECTOR of X + Lin{v}
  such that
A1: v = y and
A2: not v in X;
  let r be Real;
    reconsider W1 = X as Subspace of X + Lin{v} by RLSUB_2:11;
   defpred P[VECTOR of X + Lin{v},Real] means
    for x being VECTOR of X,
     s being Real st ($1 |-- (W1,Lin{y}))`1 = x & ($1 |-- (W1,Lin{y}))`2 = s*v
    holds $2 = fi.x + s*r;
A3:  now let e be Element of X + Lin{v};
      consider x being VECTOR of X, s being Real such that
A4:     e |-- (W1,Lin{y}) = [x,s*v]
       by A1,A2,Th28;
     take u = fi.x + s*r;
     thus P[e,u]
     proof
      let x' be VECTOR of X, t be Real
      such that
A5:     (e |-- (W1,Lin{y}))`1 = x' & (e |-- (W1,Lin{y}))`2 = t*v;
A6:    v <> 0.V by A2,RLSUB_1:25;
        s*v = t * v by A4,A5,MCART_1:7;
      then t = s by A6,RLVECT_1:51;
      hence u = fi.x' + t*r by A4,A5,MCART_1:7;
     end;
    end;
   consider f being Function of the carrier of X + Lin{v},REAL such that
A7:  for e being VECTOR of X + Lin{v} holds P[e,f.e] from FuncExD(A3);
A8: dom fi = the carrier of X by FUNCT_2:def 1;
A9: now let a be set; assume
    a in dom fi;
     then reconsider x = a as VECTOR of X by FUNCT_2:def 1;
       the carrier of X c= the carrier of X + Lin{v} by Th14;
     then reconsider v1 = x as VECTOR of X + Lin{v} by TARSKI:def 3;
       v1 in X by RLVECT_1:def 1;
   then (v1 |-- (W1,Lin{y})) = [v1,0.V] by A1,A2,Th26
                         .= [v1,0*v] by RLVECT_1:23;
then A10:   (v1 |-- (W1,Lin{y}))`1 = x & (v1 |-- (W1,Lin{y}))`2 = 0*v by
MCART_1:7;
    thus fi.a = fi.x + 0*r
         .= f.a by A7,A10;
   end;
   reconsider f as Functional of X + Lin{v};
A11: f is additive
    proof let v1,v2 be VECTOR of X + Lin{v};
      consider x1 being VECTOR of X, s1 being Real such that
A12:    v1 |-- (W1,Lin{y}) = [x1,s1*v] by A1,A2,Th28;
      consider x2 being VECTOR of X, s2 being Real such that
A13:    v2 |-- (W1,Lin{y}) = [x2,s2*v] by A1,A2,Th28;
A14:       (v1 |-- (W1,Lin{y}))`1 = x1 & (v1 |-- (W1,Lin{y}))`2 = s1*v
         by A12,MCART_1:7;
A15:       (v2 |-- (W1,Lin{y}))`1 = x2 & (v2 |-- (W1,Lin{y}))`2 = s2*v
         by A13,MCART_1:7;
        v1 + v2 |-- (W1,Lin{y}) = [x1 +x2,(s1+s2)*v] by A1,A2,A12,A13,Th29;
      then (v1 + v2 |-- (W1,Lin{y}))`1 = x1 + x2 &
       (v1 + v2 |-- (W1,Lin{y}))`2 = (s1+ s2)*v by MCART_1:7;
     hence f.(v1+v2) = fi.(x1 + x2) + (s1 + s2)*r by A7
           .= fi.(x1 + x2) + (s1*r + s2*r) by XCMPLX_1:8
           .= fi.(x1) + fi.(x2) + (s1*r + s2*r) by Def4
           .= fi.(x1) + fi.(x2) + s1*r + s2*r by XCMPLX_1:1
           .= fi.(x1) + s1*r + fi.(x2) + s2*r by XCMPLX_1:1
           .= fi.(x1) + s1*r + (fi.(x2) + s2*r) by XCMPLX_1:1
           .= f.v1 + (fi.(x2) + s2*r) by A7,A14
           .= f.v1+f.v2 by A7,A15;
    end;
     f is homogeneous
    proof let v0 be VECTOR of X + Lin{v}, t be Real;
      consider x0 being VECTOR of X, s0 being Real such that
A16:    v0 |-- (W1,Lin{y}) = [x0,s0*v] by A1,A2,Th28;
A17:       (v0 |-- (W1,Lin{y}))`1 = x0 & (v0 |-- (W1,Lin{y}))`2 = s0*v
         by A16,MCART_1:7;
        t*v0 |-- (W1,Lin{y}) = [t*x0,t*s0*v] by A1,A2,A16,Th30;
      then (t*v0 |-- (W1,Lin{y}))`1 = t*x0 &
       (t*v0 |-- (W1,Lin{y}))`2 = t*s0*v by MCART_1:7;
     hence f.(t*v0) = fi.(t*x0) + t*s0*r by A7
        .= t*(fi.x0) + t*s0*r by Def5
        .= t*(fi.x0) + t*(s0*r) by XCMPLX_1:4
        .= t*(fi.x0 + s0*r) by XCMPLX_1:8
        .= t*f.v0 by A7,A17;
    end;
   then reconsider f as linear-Functional of X + Lin{v} by A11;
  take f;
     dom f = the carrier of X + Lin{v} by FUNCT_2:def 1;
   then dom fi c= dom f by A8,Th14;
   then dom fi = dom f /\ the carrier of X by A8,XBOOLE_1:28;
  hence f|the carrier of X=fi by A9,FUNCT_1:68;
A18:  y |-- (W1,Lin{y}) = [0.W1,y] by A1,A2,Th25;
then A19: (y |-- (W1,Lin{y}))`1 = 0.X by MCART_1:7;
     (y |-- (W1,Lin{y}))`2 = y by A18,MCART_1:7 .= 1*v by A1,RLVECT_1:def 9;
 hence f.y = fi.(0.X) + 1*r by A7,A19 .= 0 + 1*r by Th33
     .= r;
end;

begin :: Hahn Banach Theorem

Lm1:
 for X being Subspace of V, v being VECTOR of V
  st not v in the carrier of X
 for q being Banach-Functional of V, fi being linear-Functional of X
  st for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v
  ex psi being linear-Functional of X + Lin{v} st
    psi|the carrier of X=fi &
  for x being VECTOR of X + Lin{v}, v being VECTOR of V st x = v
    holds psi.x <= q.v
 proof let X be Subspace of V, v be VECTOR of V such that
A1: not v in the carrier of X;
  let q be Banach-Functional of V, fi be linear-Functional of X such that
A2: for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v;
A3: the carrier of Lin{v} c= the carrier of Lin{v} + X by Th14;
    v in Lin{v} by RLVECT_4:12;
  then v in the carrier of Lin{v} by RLVECT_1:def 1;
  then reconsider x0 = v as VECTOR of X + Lin{v} by A3,RLSUB_2:9;
  consider x1 being VECTOR of X;
  reconsider v1 = x1 as VECTOR of V by RLSUB_1:18;
  set
   A = { fi.x - q.(y - v) where x is VECTOR of X, y is VECTOR of V : x = y },
   B = { fi.x + q.(v - y) where x is VECTOR of X, y is VECTOR of V : x = y };
A4: A c= REAL
   proof let a be set;
    assume a in A;
     then ex x being VECTOR of X, y being VECTOR of V st
     a = fi.x - q.(y - v) & x = y;
    hence a in REAL;
   end;
A5: B c= REAL
   proof let b be set;
    assume b in B;
     then ex x being VECTOR of X, y being VECTOR of V st
     b = fi.x + q.(v - y) & x = y;
    hence b in REAL;
   end;
A6: now let x1,x2 be VECTOR of X, y1,y2 be VECTOR of V; assume
     x1 = y1 & x2 = y2;
then x1 - x2 = y1 - y2 by RLSUB_1:24;
      then fi.(x1 - x2) <= q.(y1 - y2) by A2;
then A7:    fi.x1 - fi.x2 <= q.(y1 - y2) by Th32;
A8:    y1 + -v = y1 - v & v + -y2 = v - y2 by RLVECT_1:def 11;
        y1 - y2 = y1 + -y2 by RLVECT_1:def 11
        .= y1 + 0.V + -y2 by RLVECT_1:10
        .= y1 + (-v + v) + -y2 by RLVECT_1:16
        .= y1 - v + v + -y2 by A8,RLVECT_1:def 6
        .= y1 - v + (v - y2) by A8,RLVECT_1:def 6;
      then q.(y1 - y2) <= q.(y1 - v) + q.(v - y2) by Def3;
      then fi.x1 - fi.x2 <= q.(v - y2) + q.(y1 - v) by A7,AXIOMS:22;
      then fi.x1 <= fi.x2 + (q.(v - y2) + q.(y1 - v)) by REAL_1:86;
      then fi.x1 <= fi.x2 + q.(v - y2) + q.(y1 - v) by XCMPLX_1:1;
     hence fi.x1 - q.(y1 - v) <= fi.x2 + q.(v - y2) by REAL_1:86;
    end;
A9: now let a,b be R_eal;
     assume a in A;
     then consider x1 being VECTOR of X, y1 being VECTOR of V such that
A10:   a = fi.x1 - q.(y1 - v) & x1 = y1;
     assume b in B;
     then consider x2 being VECTOR of X, y2 being VECTOR of V such that
A11:   b = fi.x2 + q.(v - y2) & x2 = y2;
        fi.x1 - q.(y1 - v) <= fi.x2 + q.(v - y2) by A6,A10,A11;
     hence a <=' b by A10,A11,SUPINF_1:def 7;
    end;
     fi.x1 - q.(v1 - v) in A & fi.x1 + q.(v - v1) in B;
  then reconsider A, B as non empty Subset of ExtREAL by A4,A5,XBOOLE_1:1;
A12: sup A <=' inf B by A9,Th10;
A13: A <> {-infty}
    proof consider x being Element of A;
     assume A = {-infty};
      then x = -infty & x in REAL by A4,TARSKI:def 1,def 3;
     hence contradiction by SUPINF_1:6;
    end;
  A is bounded_above
   proof consider b being Element of B;
     reconsider b as Element of ExtREAL;
       b is majorant of A
      proof let x be R_eal; thus thesis by A9; end;
     then reconsider b as majorant of A;
    take b;
    thus thesis by A5,TARSKI:def 3;
   end;
  then reconsider r = sup A as Real by A13,SUPINF_1:83;
    not v in X by A1,RLVECT_1:def 1;
  then consider psi being linear-Functional of X + Lin{v}
   such that
A14: psi|the carrier of X=fi and
A15: psi.x0 = r by Th34;
  take psi;
  thus psi|the carrier of X=fi by A14;
  let y be VECTOR of X + Lin{v}, u be VECTOR of V such that
A16: y = u;
     y in X + Lin{v} by RLVECT_1:def 1;
   then consider y1,y2' being VECTOR of V such that
A17: y1 in X & y2' in Lin{v} and
A18: y = y1 + y2' by RLSUB_2:5;
   consider t being Real such that
A19: y2' = t * v by A17,RLVECT_4:11;

     y1 in X + Lin{v} by A17,RLSUB_2:6;
   then reconsider x = y1 as VECTOR of X + Lin{v} by RLVECT_1:def 1;
   reconsider x1 = x as VECTOR of X by A17,RLVECT_1:def 1;

  per cases;
   suppose t = 0;
    then y2' = 0.V by A19,RLVECT_1:23;
    then A20:   y = x1 by A18,RLVECT_1:10;
    then fi.x1 <= q.u by A2,A16;
    hence thesis by A14,A20,FUNCT_1:72;
   suppose
A21:  t > 0;
     y2' in X + Lin{v} by A17,RLSUB_2:6;
   then reconsider y2 = y2' as VECTOR of X + Lin{v} by RLVECT_1:def 1;
     y2 = t*x0 by A19,RLSUB_1:22;
   then A22: y = x + t*x0 by A18,RLSUB_1:21;
A23: t" > 0 by A21,REAL_1:72;
A24: (-t)"*x1 = (-t)"*y1 by RLSUB_1:22;
 then (-t)"*x1 = (-t)"*x by RLSUB_1:22;
then A25: fi.((-t)"*x1) = psi.((-t)"*x) by A14,FUNCT_1:72;
A26:  t*(t"*(q.u)) = t*t"*(q.u) by XCMPLX_1:4
      .= 1*(q.u) by A21,XCMPLX_0:def 7
      .= q.u;
A27:  psi.(x + t*x0) = psi.x + psi.(t*x0) by Def4
      .= 1*(psi.x + t*psi.x0) by Def5
      .= t*t"*(psi.x + t*psi.x0) by A21,XCMPLX_0:def 7
      .= t*(t"*(psi.x + t*psi.x0)) by XCMPLX_1:4
      .= t*(t"*(psi.x) + t"*(t*psi.x0)) by XCMPLX_1:8
      .= t*(t"*(psi.x) + t"*t*psi.x0) by XCMPLX_1:4
      .= t*(t"*(psi.x) + 1*psi.x0) by A21,XCMPLX_0:def 7
      .= t*(psi.(t"*x) + r) by A15,Def5;

    reconsider b = psi.((-t)"*x) + q.(v - (-t)"*y1) as R_eal by SUPINF_1:def 6,
XBOOLE_0:def 2;
      psi.((-t)"*x) + q.(v - (-t)"*y1) in B by A24,A25;
    then inf B <=' b by SUPINF_1:79;
    then sup A <=' b by A12,SUPINF_1:29;
    then psi.((-t)"*x) + q.(v - (-t)"*y1) >= r by Th12;
    then psi.((-t)"*x) >= r - q.(v - (-t)"*y1) by REAL_1:86;
then A28: - psi.((-t)"*x) <= - (r - q.(v - (-t)"*y1)) by REAL_1:50;
A29: - psi.((-t)"*x) = (-1)*psi.((-t)"*x) by XCMPLX_1:180
       .= psi.((-1)*((-t)"*x)) by Def5
       .= psi.((-1)*(-t)"*x) by RLVECT_1:def 9
       .= psi.((-1)*(-t")*x) by XCMPLX_1:224
       .= psi.(1*t"*x) by XCMPLX_1:177
       .= psi.(t"*x);
       v - (-t)"*y1 = v - (-t")*y1 by XCMPLX_1:224
         .= v - -t"*y1 by RLVECT_4:6
         .= v + - -t"*y1 by RLVECT_1:def 11
         .= v + t"*y1 by RLVECT_1:30;
then A30: psi.(t"*x) <= q.(v + t"*y1) - r by A28,A29,XCMPLX_1:143;
      t"*u = t"*y1 + t"*y2' by A16,A18,RLVECT_1:def 9
      .= t"*y1 + t"*t*v by A19,RLVECT_1:def 9
      .= t"*y1 + 1*v by A21,XCMPLX_0:def 7
      .= v + t"*y1 by RLVECT_1:def 9;
    then psi.(t"*x) + r <= q.(t"*u) by A30,REAL_1:84;
   then psi.(t"*x) + r <= t"*(q.u) by A23,Def6;
  hence psi.y <= q.u by A21,A22,A26,A27,AXIOMS:25;
   suppose t < 0;
then A31:  -t > 0 by REAL_1:66;
A32:  -y2' = (-t)*v by A19,RLVECT_4:6;
A33:  -y2' in Lin{v} by A17,RLSUB_1:30;
A34:  y = y1 + --y2' by A18,RLVECT_1:30 .= y1 - -y2' by RLVECT_1:def 11;
     -y2' in X + Lin{v} by A33,RLSUB_2:6;
   then reconsider y2 = -y2' as VECTOR of X + Lin{v} by RLVECT_1:def 1;
     y2 = (-t)*x0 by A32,RLSUB_1:22;
   then A35: y = x - (-t)*x0 by A34,RLSUB_1:24;
A36: (-t)" > 0 by A31,REAL_1:72;
A37: (-t)"*x1 = (-t)"*y1 by RLSUB_1:22;
 then (-t)"*x1 = (-t)"*x by RLSUB_1:22;
then A38: fi.((-t)"*x1) = psi.((-t)"*x) by A14,FUNCT_1:72;
A39:  (-t)*((-t)"*(q.u)) = (-t)*(-t)"*(q.u) by XCMPLX_1:4
      .= 1*(q.u) by A31,XCMPLX_0:def 7
      .= q.u;
A40:  psi.(x - (-t)*x0) = psi.x - psi.((-t)*x0) by Th32
      .= 1*(psi.x - (-t)*psi.x0) by Def5
      .= (-t)*(-t)"*(psi.x - (-t)*psi.x0) by A31,XCMPLX_0:def 7
      .= (-t)*((-t)"*(psi.x - (-t)*psi.x0)) by XCMPLX_1:4
      .= (-t)*((-t)"*(psi.x) - (-t)"*((-t)*psi.x0)) by XCMPLX_1:40
      .= (-t)*((-t)"*(psi.x) - (-t)"*(-t)*psi.x0) by XCMPLX_1:4
      .= (-t)*((-t)"*(psi.x) - 1*psi.x0) by A31,XCMPLX_0:def 7
      .= (-t)*(psi.((-t)"*x) - r) by A15,Def5;
    reconsider a = psi.((-t)"*x) - q.((-t)"*y1 - v) as R_eal by SUPINF_1:def 6,
XBOOLE_0:def 2;
      psi.((-t)"*x) - q.((-t)"*y1 - v) in A by A37,A38;
    then a <=' sup A by SUPINF_1:76;
    then psi.((-t)"*x) - q.((-t)"*y1 - v) <= r by Th12;
then A41: psi.((-t)"*x) <= r + q.((-t)"*y1 - v) by REAL_1:86;
      (-t)"*u = (-t)"*y1 - (-t)"*-y2' by A16,A34,RLVECT_1:48
      .= (-t)"*y1 - (-t)"*(-t)*v by A32,RLVECT_1:def 9
      .= (-t)"*y1 - 1*v by A31,XCMPLX_0:def 7
      .= (-t)"*y1 - v by RLVECT_1:def 9;
    then psi.((-t)"*x) - r <= q.((-t)"*u) by A41,REAL_1:86;
    then psi.((-t)"*x) - r <= (-t)"*(q.u) by A36,Def6;
  hence psi.y <= q.u by A31,A35,A39,A40,AXIOMS:25;
 end;

Lm2:
 the RLSStruct of V is RealLinearSpace
  proof
A1: 0.the RLSStruct of V = the Zero of V by RLVECT_1:def 2
       .= 0.V by RLVECT_1:def 2;
A2: now let v',w' be VECTOR of V, v,w be VECTOR of the RLSStruct of V; assume
A3:   v = v' & w = w';
     hence v + w = (the add of V).[v',w'] by RLVECT_1:def 3
        .= v' + w' by RLVECT_1:def 3;
     let r be Real;
     thus r *v = (the Mult of V).[r,v'] by A3,RLVECT_1:def 4
            .= r* v' by RLVECT_1:def 4;
    end;
      the RLSStruct of V is Abelian
     add-associative right_zeroed right_complementable RealLinearSpace-like
     proof
      hereby let v,w be VECTOR of the RLSStruct of V;
        reconsider v' = v, w' = w as VECTOR of V;
       thus v + w = w' + v' by A2
          .= w + v by A2;
      end;
      hereby let u,v,w be VECTOR of the RLSStruct of V;
        reconsider u' = u, v' = v, w' = w as VECTOR of V;
A4:      v + w = v' + w' by A2;
          u + v = u' + v' by A2;
       hence (u + v) + w = (u' + v') + w' by A2
           .= u' + (v' + w') by RLVECT_1:def 6
           .= u + (v + w) by A2,A4;
      end;
      hereby let v be VECTOR of the RLSStruct of V;
        reconsider v' = v as VECTOR of V;
       thus v + 0.the RLSStruct of V = v' + 0.V by A1,A2
          .= v by RLVECT_1:10;
      end;
      hereby let v be VECTOR of the RLSStruct of V;
        reconsider v' = v as VECTOR of V;
        consider w' being VECTOR of V such that
A5:       v' + w' = 0.V by RLVECT_1:def 8;
        reconsider w = w' as VECTOR of the RLSStruct of V;
       take w;
       thus v + w = 0.the RLSStruct of V by A1,A2,A5;
      end;
      hereby let a be Real, v,w be VECTOR of the RLSStruct of V;
        reconsider v' = v, w' = w as VECTOR of V;
A6:     a * v = a * v' & a * w = a * w' by A2;
          v + w = v' + w' by A2;
       hence a * (v + w) = a * (v' + w') by A2
           .= a * v' + a * w' by RLVECT_1:def 9
           .= a * v + a * w by A2,A6;
      end;
      hereby let a,b be Real, v be VECTOR of the RLSStruct of V;
        reconsider v' = v as VECTOR of V;
A7:      a * v = a * v' & b * v = b * v' by A2;
       thus (a + b) * v = (a + b) * v' by A2
         .= a * v' + b * v' by RLVECT_1:def 9
         .= a * v + b * v by A2,A7;
      end;
      hereby let a,b be Real, v be VECTOR of the RLSStruct of V;
        reconsider v' = v as VECTOR of V;
A8:      b * v = b * v' by A2;
       thus (a * b) * v = (a * b) * v' by A2
          .= a * (b * v') by RLVECT_1:def 9
          .= a * (b * v) by A2,A8;
      end;
      let v be VECTOR of the RLSStruct of V;
       reconsider v' = v as VECTOR of V;
      thus 1 * v = 1 * v' by A2
         .= v by RLVECT_1:def 9;
     end;
   hence thesis;
  end;

Lm3: for V',W' being RealLinearSpace st V' = the RLSStruct of V
 for W being Subspace of V st W' = the RLSStruct of W
  holds W' is Subspace of V'
  proof let V',W' be RealLinearSpace such that
A1: V' = the RLSStruct of V;
   let W be Subspace of V; assume
A2: W' = the RLSStruct of W;
   hence the carrier of W' c= the carrier of V' by A1,RLSUB_1:def 2;
   thus the Zero of W' = the Zero of V' by A1,A2,RLSUB_1:def 2;
   thus the add of W'
      = (the add of V') | [:the carrier of W',the carrier of W':]
       by A1,A2,RLSUB_1:def 2;
   thus the Mult of W' = (the Mult of V') | [:REAL, the carrier of W':]
      by A1,A2,RLSUB_1:def 2;
  end;

Lm4: for V' being RealLinearSpace st V' = the RLSStruct of V
 for f being linear-Functional of V'
  holds f is linear-Functional of V
 proof let V' be RealLinearSpace such that
A1: V' = the RLSStruct of V;
  let f be linear-Functional of V';
   reconsider f' = f as Functional of V by A1;
A2: f' is additive
    proof let x,y be VECTOR of V;
      reconsider x' = x, y' = y as VECTOR of V' by A1;
     thus f'.(x+y) = f.((the add of V').[x',y']) by A1,RLVECT_1:def 3
        .= f.(x'+y') by RLVECT_1:def 3
        .= f'.x+f'.y by Def4;
    end;
     f' is homogeneous
    proof let x be VECTOR of V, r be Real;
      reconsider x' = x as VECTOR of V' by A1;
     thus f'.(r*x) = f.((the Mult of V').[r,x']) by A1,RLVECT_1:def 4
        .= f'.(r*x') by RLVECT_1:def 4
        .= r*f'.x by Def5;
    end;
  hence f is linear-Functional of V by A2;
 end;

Lm5: for V' being RealLinearSpace st V' = the RLSStruct of V
 for f being linear-Functional of V
  holds f is linear-Functional of V'
 proof let V' be RealLinearSpace such that
A1: V' = the RLSStruct of V;
  let f be linear-Functional of V;
   reconsider f' = f as Functional of V' by A1;
A2: f' is additive
    proof let x,y be VECTOR of V';
      reconsider x' = x, y' = y as VECTOR of V by A1;
     thus f'.(x+y) = f.((the add of V).[x',y']) by A1,RLVECT_1:def 3
        .= f.(x'+y') by RLVECT_1:def 3
        .= f'.x+f'.y by Def4;
    end;
     f' is homogeneous
    proof let x be VECTOR of V', r be Real;
      reconsider x' = x as VECTOR of V by A1;
     thus f'.(r*x) = f.((the Mult of V).[r,x']) by A1,RLVECT_1:def 4
        .= f'.(r*x') by RLVECT_1:def 4
        .= r*f'.x by Def5;
    end;
  hence f is linear-Functional of V' by A2;
 end;

theorem Th35:
 for V being RealLinearSpace, X being Subspace of V,
  q being Banach-Functional of V, fi being linear-Functional of X
  st for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v
  ex psi being linear-Functional of V st
  psi|the carrier of X=fi &
  for x being VECTOR of V holds psi.x <= q.x
  proof let V be RealLinearSpace, X be Subspace of V,
    q be Banach-Functional of V, fi be linear-Functional of X;
   assume
A1:  for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= q.v;
   defpred P[Element of PFuncs(the carrier of V,REAL)] means
   ex Y being Subspace of V st
      the carrier of X c= the carrier of Y & $1|the carrier of X = fi &
     ex f' being linear-Functional of Y st $1 = f' &
      for y being VECTOR of Y, v being VECTOR of V st y = v holds
       f'.y <= q.v;
   reconsider A = { f where f is Element of
   PFuncs(the carrier of V,REAL): P[f]}
   as Subset of PFuncs(the carrier of V,REAL) from SubsetD;
    the carrier of X c= the carrier of V by RLSUB_1:def 2;
    then fi in PFuncs(the carrier of X,REAL) &
     PFuncs(the carrier of X,REAL) c= PFuncs(the carrier of V,REAL)
     by PARTFUN1:119,125;
    then reconsider fi' = fi as Element of PFuncs(the carrier of V,REAL);
     fi'|the carrier of X = fi by FUNCT_2:40;
    then fi in A by A1;
    then reconsider A as non empty Subset of PFuncs(the carrier of V,REAL);
    defpred P[set, set] means $1 c= $2;
A2:  for x being Element of A holds P[x,x];
A3:  for x,y being Element of A st P[x,y] & P[y,x] holds x = y by XBOOLE_0:def
10;
A4:  for x,y,z being Element of A st P[x,y] & P[y,z] holds P[x,z] by XBOOLE_1:1
;
A5:  now let B be set such that
A6:   B c= A and
A7:   for x,y being Element of A st x in B & y in B
       holds P[x,y] or P[y,x];
     per cases;
     suppose
A8:    B = {};
      consider u being Element of A;
     take u;
     let x be Element of A;
     assume x in B;
     hence P[x,u] by A8;
     suppose
A9:    B <> {};
      A10: B is Subset of PFuncs(the carrier of V,REAL) by A6,XBOOLE_1:1;
      then reconsider E = B as non empty functional set by A9,Th4;
        B is c=-linear
      proof let x,y be set;
       assume x in B & y in B;
        hence x c= y or y c= x by A6,A7;
      end;
      then reconsider u = union B as Element of PFuncs(the carrier of V,REAL)
        by A10,Th13;
      consider t being Element of B;
A11:   now let t be set;
        assume t in A;
       then consider f being Element of PFuncs(the carrier of V,REAL)
       such that
  A12:    t = f and
  A13:   ex Y being Subspace of V st
       the carrier of X c= the carrier of Y & f|the carrier of X = fi &
       ex f' being linear-Functional of Y st f = f' &
        for y being VECTOR of Y, v being VECTOR of V st y = v holds
         f'.y <= q.v;
       consider Y being Subspace of V such that
  A14:    the carrier of X c= the carrier of Y and
  A15:    f|the carrier of X = fi and
  A16:    ex f' being linear-Functional of Y st f = f' &
         for y being VECTOR of Y, v being VECTOR of V st y = v holds
          f'.y <= q.v by A13;
       consider f' being linear-Functional of Y such that
  A17:    f = f' and
A18:     for y being VECTOR of Y, v being VECTOR of V st y = v holds
          f'.y <= q.v by A16;
       take Y;
        reconsider f = f' as linear-Functional of Y;
       take f;
       thus t = f by A12,A17;
       thus the carrier of X c= the carrier of Y by A14;
       thus f|the carrier of X = fi by A15,A17;
       thus for y being VECTOR of Y, v being VECTOR of V st y = v holds
          f.y <= q.v by A18;
      end;
        t in A by A6,A9,TARSKI:def 3;
      then consider Y being Subspace of V, f' being linear-Functional of Y
      such that
A19:    t = f' and
         the carrier of X c= the carrier of Y &
       f'|the carrier of X = fi &
       for y being VECTOR of Y, v being VECTOR of V st y = v holds
          f'.y <= q.v by A11;
      A20: dom f' = the carrier of Y by FUNCT_2:def 1;
        t c= u by A9,ZFMISC_1:92;
then A21:   u <> {} by A19,A20,RELAT_1:60,XBOOLE_1:3;
      set K = { dom g where g is Element of E: not contradiction };
A22:   dom u = union K by Th5;
        now let x be set;
       assume x in { dom g where g is Element of E: not contradiction };
        then consider g being Element of E such that
A23:      dom g = x;
          g in A by A6,TARSKI:def 3;
        then consider Y being Subspace of V, f' being linear-Functional of Y
        such that
A24:      g = f' and
           the carrier of X c= the carrier of Y &
         f'|the carrier of X = fi &
         for y being VECTOR of Y, v being VECTOR of V st y = v holds
            f'.y <= q.v by A11;
          dom g = the carrier of Y by A24,FUNCT_2:def 1;
       hence x c= the carrier of V by A23,RLSUB_1:def 2;
      end;
      then reconsider D = dom u as non empty Subset of V
        by A21,A22,RELAT_1:64,ZFMISC_1:94;
        D is lineary-closed
       proof
        hereby let v,u be Element of V; assume
A25:       v in D & u in D;
          then consider D1 being set such that
A26:        v in D1 & D1 in K by A22,TARSKI:def 4;
          consider g1 being Element of E such that
A27:        D1 = dom g1 by A26;
          consider D2 being set such that
A28:        u in D2 & D2 in K by A22,A25,TARSKI:def 4;
          consider g2 being Element of E such that
A29:        D2 = dom g2 by A28;
            g1 in A & g2 in A by A6,TARSKI:def 3;
          then g1 c= g2 or g2 c= g1 by A7;
          then consider g being Element of E such that
A30:        g1 c= g & g2 c= g;
          A31: D1 c= dom g & D2 c= dom g by A27,A29,A30,RELAT_1:25;
            g in A by A6,TARSKI:def 3;
        then consider Y being Subspace of V, f' being linear-Functional of Y
        such that
A32:      g = f' and
           the carrier of X c= the carrier of Y &
         f'|the carrier of X = fi &
         for y being VECTOR of Y, v being VECTOR of V st y = v holds
            f'.y <= q.v by A11;
A33:       dom g = the carrier of Y by A32,FUNCT_2:def 1;
          then v in Y & u in Y by A26,A28,A31,RLVECT_1:def 1;
          then v + u in Y by RLSUB_1:28;
          then v + u in dom g & dom g in K by A33,RLVECT_1:def 1;
         hence v + u in D by A22,TARSKI:def 4;
        end;
        let a be Real, v be Element of V; assume
         v in D;
         then consider D1 being set such that
A34:        v in D1 & D1 in K by A22,TARSKI:def 4;
         consider g being Element of E such that
A35:        D1 = dom g by A34;
           g in A by A6,TARSKI:def 3;
       then consider Y being Subspace of V, f' being linear-Functional of Y
       such that
A36:      g = f' and
          the carrier of X c= the carrier of Y &
        f'|the carrier of X = fi &
        for y being VECTOR of Y, v being VECTOR of V st y = v holds
           f'.y <= q.v by A11;
A37:       dom g = the carrier of Y by A36,FUNCT_2:def 1;
         then v in Y by A34,A35,RLVECT_1:def 1;
         then a * v in Y by RLSUB_1:29;
         then a * v in dom g & dom g in K by A37,RLVECT_1:def 1;
        hence a * v in D by A22,TARSKI:def 4;
       end;
      then consider Y being strict Subspace of V such that
A38:    the carrier of Y = dom u by RLSUB_1:43;
      consider t being Element of dom u;
      consider XX being set such that
A39:    t in XX & XX in K by A22,A38,TARSKI:def 4;
      consider gg being Element of E such that
A40:    XX = dom gg by A39;
        gg in A by A6,TARSKI:def 3;
      then consider YY being Subspace of V, ff being linear-Functional of YY
      such that
A41:   gg = ff and
A42:   the carrier of X c= the carrier of YY and
A43:   ff|the carrier of X = fi and
         for y being VECTOR of YY, v being VECTOR of V st y = v holds
          ff.y <= q.v by A11;
A44:  XX = the carrier of YY by A40,A41,FUNCT_2:def 1;
        XX c= dom u by A22,A39,ZFMISC_1:92;
      then A45:    the carrier of X c= the carrier of Y by A38,A42,A44,XBOOLE_1
:1;
A46:   the carrier of X c= dom ff by A42,FUNCT_2:def 1;
        ff c= u by A41,ZFMISC_1:92;
      then A47:   u|the carrier of X = fi by A43,A46,Th2;
        ex f being Function st u = f & dom f c= the carrier of V &
         rng f c= REAL by PARTFUN1:def 5;
      then reconsider f' = u as Function of the carrier of Y, REAL
       by A38,FUNCT_2:def 1,RELSET_1:11;
      reconsider f' as Functional of Y;
A48:  f' is additive
       proof let x,y be VECTOR of Y;
         consider XXx being set such that
    A49:    x in XXx & XXx in K by A22,A38,TARSKI:def 4;
         consider ggx being Element of E such that
    A50:    XXx = dom ggx by A49;
A51:      ggx in A by A6,TARSKI:def 3;
         consider XXy being set such that
    A52:    y in XXy & XXy in K by A22,A38,TARSKI:def 4;
         consider ggy being Element of E such that
    A53:    XXy = dom ggy by A52;
       ggy in A by A6,TARSKI:def 3;
         then ggx c= ggy or ggy c= ggx by A7,A51;
         then consider gg being Element of E such that
A54:       gg = ggx or gg = ggy and
A55:       ggx c= gg & ggy c= gg;
            gg in A by A6,TARSKI:def 3;
        then consider YY being Subspace of V,
                      ff being linear-Functional of YY such that
A56:      gg = ff and
           the carrier of X c= the carrier of YY &
         ff|the carrier of X = fi &
         for y being VECTOR of YY, v being VECTOR of V st y = v holds
            ff.y <= q.v by A11;
A57:      dom ff = the carrier of YY by FUNCT_2:def 1;
           dom ggx c= dom gg & dom ggy c= dom gg by A55,RELAT_1:25;
         then reconsider x' = x, y' = y as VECTOR of YY by A49,A50,A52,A53,A56,
FUNCT_2:def 1;
A58:      ff c= f' by A56,ZFMISC_1:92;
           the carrier of YY c= the carrier of Y
          by A22,A38,A49,A50,A52,A53,A54,A56,A57,ZFMISC_1:92;
         then YY is Subspace of Y by RLSUB_1:36;
         then x + y = x' + y' by RLSUB_1:21;
        hence f'.(x+y) = ff.(x'+y') by A57,A58,GRFUNC_1:8
               .= ff.x' + ff.y' by Def4
               .= f'.x+ff.y' by A57,A58,GRFUNC_1:8
               .= f'.x+f'.y by A57,A58,GRFUNC_1:8;
       end;
        f' is homogeneous
       proof let x be VECTOR of Y, r be Real;
         consider XX being set such that
A59:        x in XX & XX in K by A22,A38,TARSKI:def 4;
         consider gg being Element of E such that
A60:       XX = dom gg by A59;
        gg in A by A6,TARSKI:def 3;
         then consider YY being Subspace of V,
              ff being linear-Functional of YY such that
A61:     gg = ff and
       the carrier of X c= the carrier of YY and
       ff|the carrier of X = fi and
            for y being VECTOR of YY, v being VECTOR of V st y = v holds
             ff.y <= q.v by A11;
A62:      dom ff = the carrier of YY by FUNCT_2:def 1;
         reconsider x' = x as VECTOR of YY by A59,A60,A61,FUNCT_2:def 1;
A63:      ff c= f' by A61,ZFMISC_1:92;
           the carrier of YY c= the carrier of Y
          by A22,A38,A59,A60,A61,A62,ZFMISC_1:92;
         then YY is Subspace of Y by RLSUB_1:36;
         then r*x = r*x' by RLSUB_1:22;
        hence f'.(r*x) = ff.(r*x') by A62,A63,GRFUNC_1:8
           .= r*ff.x' by Def5
           .= r*f'.x by A62,A63,GRFUNC_1:8;
       end;
      then reconsider f' as linear-Functional of Y by A48;
        now let y be VECTOR of Y, v be VECTOR of V such that
A64:     y = v;
        consider XX being set such that
A65:        y in XX & XX in K by A22,A38,TARSKI:def 4;
        consider gg being Element of E such that
A66:       XX = dom gg by A65;
        gg in A by A6,TARSKI:def 3;
        then consider YY being Subspace of V,
             ff being linear-Functional of YY such that
A67:     gg = ff and
       the carrier of X c= the carrier of YY and
       ff|the carrier of X = fi and
A68:     for y being VECTOR of YY, v being VECTOR of V st y = v holds
            ff.y <= q.v by A11;
A69:      dom ff = the carrier of YY by FUNCT_2:def 1;
         reconsider y' = y as VECTOR of YY by A65,A66,A67,FUNCT_2:def 1;
A70:      ff c= f' by A67,ZFMISC_1:92;
          ff.y' <= q.v by A64,A68;
       hence f'.y <= q.v by A69,A70,GRFUNC_1:8;
      end;
      then u in A by A45,A47;
      then reconsider u as Element of A;
     take u;
     let x be Element of A;
     assume x in B;
     hence P[x,u] by ZFMISC_1:92;
    end;
    consider psi being Element of A such that
A71:   for chi being Element of A st psi <> chi holds
        not P[psi,chi] from Zorn_Max(A2,A3,A4,A5);
   psi in A;
    then consider f being Element of PFuncs(the carrier of V,REAL) such that
A72:  f = psi and
A73:  ex Y being Subspace of V st
      the carrier of X c= the carrier of Y & f|the carrier of X = fi &
     ex f' being linear-Functional of Y st f = f' &
      for y being VECTOR of Y, v being VECTOR of V st y = v holds
       f'.y <= q.v;
    consider Y being Subspace of V such that
A74:   the carrier of X c= the carrier of Y and
A75:   f|the carrier of X = fi and
A76:   ex f' being linear-Functional of Y st f = f' &
      for y being VECTOR of Y, v being VECTOR of V st y = v holds
       f'.y <= q.v by A73;
    consider f' being linear-Functional of Y such that
A77:   f = f' and
A78:   for y being VECTOR of Y, v being VECTOR of V st y = v holds
       f'.y <= q.v by A76;
    reconsider RLSY = the RLSStruct of Y as RealLinearSpace by Lm2;
    reconsider ggh = f' as linear-Functional of RLSY by Lm5;
A79: f = ggh by A77;
    reconsider RLS = the RLSStruct of V as RealLinearSpace by Lm2;
A80: RLSY is Subspace of RLS by Lm3;
A81: now assume the RLSStruct of Y <> the RLSStruct of V;
      then the carrier of Y <> the carrier of V &
       the carrier of Y c= the carrier of V by A80,RLSUB_1:40,def 2;
       then the carrier of Y c< the carrier of V by XBOOLE_0:def 8;
      then consider v being set such that
A82:     v in the carrier of V & not v in the carrier of Y by RLSUB_2:77;
      reconsider v as VECTOR of V by A82;
      consider phi being linear-Functional of Y + Lin{v} such that
A83:    phi|the carrier of Y = f' and
A84:    for x being VECTOR of Y + Lin{v}, v being VECTOR of V st x = v
        holds phi.x <= q.v by A78,A82,Lm1;
A85:   dom phi = the carrier of Y + Lin{v} by FUNCT_2:def 1;
     the carrier of Y c= the carrier of Y + Lin{v} by Th14;
then A86:   the carrier of X c= the carrier of Y + Lin{v} by A74,XBOOLE_1:1;
A87:   dom phi c= the carrier of V by A85,RLSUB_1:def 2;
        rng phi c= REAL by RELSET_1:12;
      then reconsider phi as Element of PFuncs(the carrier of V,REAL)
        by A87,PARTFUN1:def 5;
        (the carrier of Y) /\
 the carrier of X = the carrier of X by A74,XBOOLE_1:28;
      then phi|the carrier of X
        = fi by A75,A77,A83,RELAT_1:100;
      then phi in A by A84,A86;
      then reconsider phi as Element of A;
A88:   dom f' = the carrier of Y by FUNCT_2:def 1;
A89:   the carrier of Lin{v} c= the carrier of Lin{v} + Y by Th14;
        v in Lin{v} by RLVECT_4:12;
      then v in the carrier of Lin{v} by RLVECT_1:def 1;
      then v in the carrier of Lin{v} + Y by A89;
      then A90:   phi <> psi by A72,A77,A82,A85,A88,RLSUB_2:9;
        psi c= phi by A72,A77,A83,RELAT_1:88;
     hence contradiction by A71,A90;
    end;
    then reconsider psi as linear-Functional of V by A72,A79,Lm4;
   take psi;
   thus psi|the carrier of X = fi by A72,A75;
   let x be VECTOR of V;
   thus psi.x <= q.x by A72,A76,A81;
  end;

theorem Th36:
 for V being RealNormSpace holds
  the norm of V is absolutely_homogeneous subadditive Functional of V
 proof let V be RealNormSpace;
   reconsider N = the norm of V as Functional of V;
A1: N is absolutely_homogeneous
    proof let x be VECTOR of V, r be Real;
     thus N.(r*x) = ||.r*x.|| by NORMSP_1:def 1
        .= abs(r)*||.x.|| by NORMSP_1:def 2
        .= abs(r)*N.x by NORMSP_1:def 1;
    end;
     N is subadditive
    proof let x,y be VECTOR of V;
        N.(x) = ||.x.|| & N.(y) = ||.y.|| & N.(x+y) = ||.x+y.||
 by NORMSP_1:def 1;
     hence N.(x+y) <= N.x+N.y by NORMSP_1:def 2;
    end;
  hence thesis by A1;
 end;

theorem
   for V being RealNormSpace, X being Subspace of V,
  fi being linear-Functional of X
  st for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= ||.v.||
  ex psi being linear-Functional of V st
  psi|the carrier of X=fi &
  for x being VECTOR of V holds psi.x <= ||.x.||
  proof
   let V be RealNormSpace, X be Subspace of V,
       fi be linear-Functional of X such that
A1:  for x being VECTOR of X, v being VECTOR of V st x=v holds fi.x <= ||.v.||
;
    reconsider q = the norm of V as Banach-Functional of V by Th36;
      now let x be VECTOR of X, v be VECTOR of V such that
A2:   x=v;
        q.v = ||.v.|| by NORMSP_1:def 1;
     hence fi.x <= q.v by A1,A2;
    end;
    then consider psi being linear-Functional of V such that
A3:  psi|the carrier of X=fi and
A4:  for x being VECTOR of V holds psi.x <= q.x by Th35;
   take psi;
   thus psi|the carrier of X=fi by A3;
   let x be VECTOR of V;
      q.x = ||.x.|| by NORMSP_1:def 1;
   hence psi.x <= ||.x.|| by A4;
  end;

Back to top