### The Mizar article:

### Real Functions Spaces

**by****Henryk Oryszczyszyn, and****Krzysztof Prazmowski**

- Received March 23, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: FUNCSDOM
- [ MML identifier index ]

environ vocabulary BINOP_1, FUNCT_2, FUNCT_1, QC_LANG1, RELAT_1, FUNCOP_1, VECTSP_1, RLVECT_1, ARYTM_1, LATTICES, FUNCSDOM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, DOMAIN_1, FUNCOP_1, STRUCT_0, RLVECT_1, REAL_1, VECTSP_1, FRAENKEL; constructors BINOP_1, DOMAIN_1, FUNCOP_1, REAL_1, VECTSP_1, FRAENKEL, MEMBERED, XBOOLE_0; clusters SUBSET_1, RLVECT_1, VECTSP_1, FUNCOP_1, RELSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, SUBSET, BOOLE, ARITHM; definitions RLVECT_1, STRUCT_0; theorems FUNCT_2, BINOP_1, FUNCOP_1, RLVECT_1, VECTSP_1, TARSKI, XCMPLX_0, XCMPLX_1; schemes BINOP_1, FUNCT_2; begin reserve x1,x2,z for set; reserve A,B for non empty set; definition let A,B; let F be BinOp of Funcs(A,B); let f,g be Element of Funcs(A,B); redefine func F.(f,g) -> Element of Funcs(A,B); coherence proof reconsider f,g as Element of Funcs(A,B) qua non empty set; F.(f,g) is Element of Funcs(A,B); hence thesis; end; end; definition let A,B,C,D be non empty set; let F be Function of [:C,D:],Funcs(A,B); let cd be Element of [:C,D:]; redefine func F.cd -> Element of Funcs(A,B); coherence proof F.cd is Element of Funcs(A,B); hence thesis; end; end; definition let A,B be non empty set; let f be Function of A,B; func @f -> Element of Funcs(A,B) equals :Def1: f; coherence by FUNCT_2:11; end; reserve f,g,h for Element of Funcs(A,REAL); definition let X,Z be non empty set; let F be (BinOp of X), f,g be Function of Z,X; redefine func F.:(f,g) -> Element of Funcs(Z,X); coherence proof F.:(f,g) in Funcs(Z,X) by FUNCT_2:11; hence thesis; end; end; definition let X,Z be non empty set; let F be (BinOp of X),a be Element of X,f be Function of Z,X; redefine func F[;](a,f) -> Element of Funcs(Z,X); coherence proof F[;](a,f) in Funcs(Z,X) by FUNCT_2:11; hence thesis; end; end; definition let A; func RealFuncAdd(A) -> BinOp of Funcs(A,REAL) means :Def2: for f,g being Element of Funcs(A,REAL) holds it.(f,g) = addreal.:(f,g); existence proof deffunc F(Element of Funcs(A,REAL),Element of Funcs(A,REAL)) = addreal.:($1,$2); consider F being BinOp of Funcs(A,REAL) such that A1: for x,y being Element of Funcs(A,REAL) holds F.(x,y) = F(x,y) from BinOpLambda; take F; let f,g; thus F.(f,g) =(addreal.:(f,g)) by A1; end; uniqueness proof let it1,it2 be BinOp of Funcs(A,REAL) such that A2: for f,g being Element of Funcs(A,REAL) holds it1.(f,g) = addreal.:(f,g) and A3: for f,g being Element of Funcs(A,REAL) holds it2.(f,g) = addreal.:(f,g); now let f,g be Element of Funcs(A,REAL); thus it1.(f,g) = addreal.:(f,g) by A2 .= it2.(f,g) by A3; end; hence thesis by BINOP_1:2; end; end; definition let A; func RealFuncMult(A) -> BinOp of Funcs(A,REAL) means :Def3: for f,g being Element of Funcs(A,REAL) holds it.(f,g) = multreal.:(f,g); existence proof deffunc F(Element of Funcs(A,REAL),Element of Funcs(A,REAL)) = multreal.:($1,$2); consider F being BinOp of Funcs(A,REAL) such that A1: for x,y being Element of Funcs(A,REAL) holds F.(x,y) = F(x,y) from BinOpLambda; take F; let f,g; thus F.(f,g) =(multreal.:(f,g)) by A1; end; uniqueness proof let it1,it2 be BinOp of Funcs(A,REAL) such that A2: for f,g being Element of Funcs(A,REAL) holds it1.(f,g) = multreal.:(f,g) and A3: for f,g being Element of Funcs(A,REAL) holds it2.(f,g) = multreal.:(f,g); now let f,g be Element of Funcs(A,REAL); thus it1.(f,g) = multreal.:(f,g) by A2 .=it2.(f,g) by A3; end; hence thesis by BINOP_1:2; end; end; definition let A; func RealFuncExtMult(A) -> Function of [:REAL,Funcs(A,REAL):],Funcs(A,REAL) means :Def4: for a being Real, f being (Element of Funcs(A,REAL)), x being (Element of A) holds (it.[a,f]).x = a*(f.x); existence proof deffunc F(Element of REAL,Element of Funcs(A,REAL)) = @(multreal[;]($1,$2)); consider F being Function of [:REAL,Funcs(A,REAL):],Funcs(A,REAL) such that A1: for a being Element of REAL, f being Element of Funcs(A,REAL) holds F.[a,f] = F(a,f) from Lambda2D; take F; let a be Real,f be Element of Funcs(A,REAL), x be Element of A; F.[a,f] = @(multreal[;](a,f)) by A1 .= (multreal[;](a,f)) by Def1; hence (F.[a,f]).x = multreal.(a,f.x) by FUNCOP_1:66 .= a*(f.x) by VECTSP_1:def 14; end; uniqueness proof let it1,it2 be Function of [:REAL,Funcs(A,REAL):],Funcs(A,REAL) such that A2: for a being Real, f being Element of Funcs(A,REAL), x being Element of A holds (it1.[a,f]).x = a*(f.x) and A3: for a being Real, f being Element of Funcs(A,REAL), x being Element of A holds (it2.[a,f]).x = a*(f.x); now let a be Element of REAL, f be Element of Funcs(A,REAL); now let x be Element of A; thus (it1.[a,f]).x = a*(f.x) by A2 .= (it2.[a,f]).x by A3; end; hence it1.[a,f] = it2.[a,f] by FUNCT_2:113; end; hence thesis by FUNCT_2:120; end; end; definition let A; func RealFuncZero(A) -> Element of Funcs(A,REAL) equals :Def5: A --> 0; coherence proof A -->0 is Function of A,REAL by FUNCOP_1:58; hence thesis by FUNCT_2:11; end; end; definition let A; func RealFuncUnit(A) -> Element of Funcs(A,REAL) equals :Def6: A --> 1; coherence proof A -->1 is Function of A,REAL by FUNCOP_1:58; hence thesis by FUNCT_2:11; end; end; Lm1: for x being (Element of A), f being Function of A,B holds x in dom f proof let x be (Element of A),f be Function of A,B; x in A; hence x in dom f by FUNCT_2:def 1; end; canceled 9; theorem Th10: h = (RealFuncAdd(A)).(f,g) iff for x being Element of A holds h.x = f.x + g.x proof A1: now assume A2: h = (RealFuncAdd(A)).(f,g); let x be Element of A; A3: x in dom (addreal.:(f,g)) by Lm1; thus h.x = (addreal.:(f,g)).x by A2,Def2 .= addreal.(f.x,g.x) by A3,FUNCOP_1:28 .= f.x + g.x by VECTSP_1:def 4; end; now assume A4: for x being Element of A holds h.x=f.x + g.x; now let x be Element of A; A5: x in dom (addreal.:(f,g)) by Lm1; thus ((RealFuncAdd(A)).(f,g)).x = (addreal.:(f,g)).x by Def2 .= addreal.(f.x,g.x) by A5,FUNCOP_1:28 .= f.x + g.x by VECTSP_1:def 4 .= h.x by A4; end; hence h = (RealFuncAdd(A)).(f,g) by FUNCT_2:113; end; hence thesis by A1; end; theorem Th11: h = (RealFuncMult(A)).(f,g) iff for x being Element of A holds h.x = f.x * g.x proof A1: now assume A2: h = (RealFuncMult(A)).(f,g); let x be Element of A; A3: x in dom (multreal.:(f,g)) by Lm1; thus h.x = (multreal.:(f,g)).x by A2,Def3 .= multreal.(f.x,g.x) by A3,FUNCOP_1:28 .= f.x * g.x by VECTSP_1:def 14; end; now assume A4: for x being Element of A holds h.x=f.x * g.x; now let x be Element of A; A5: x in dom (multreal.:(f,g)) by Lm1; thus ((RealFuncMult(A)).(f,g)).x = (multreal.:(f,g)).x by Def3 .= multreal.(f.x,g.x) by A5,FUNCOP_1:28 .= f.x * g.x by VECTSP_1:def 14 .= h.x by A4; end; hence h = (RealFuncMult(A)).(f,g) by FUNCT_2:113; end; hence thesis by A1; end; theorem Th12: for x being Element of A holds (RealFuncUnit(A)).x = 1 proof let x be Element of A; thus (RealFuncUnit(A)).x = (A --> 1).x by Def6 .= 1 by FUNCOP_1:13; end; theorem Th13: for x being Element of A holds (RealFuncZero(A)).x = 0 proof let x be Element of A; thus (RealFuncZero(A)).x = (A --> 0).x by Def5 .= 0 by FUNCOP_1:13; end; theorem RealFuncZero(A) <> RealFuncUnit(A) proof consider a being Element of A; (RealFuncZero(A)).a=0 & (RealFuncUnit(A)).a=1 by Th12,Th13; hence thesis; end; reserve a,b for Real; theorem Th15: h = (RealFuncExtMult(A)).[a,f] iff for x being Element of A holds h.x = a*(f.x) proof thus h = (RealFuncExtMult(A)).[a,f] implies (for x being Element of A holds h.x = a*(f.x)) by Def4; now assume A1: for x being Element of A holds h.x = a*(f.x); for x being Element of A holds h.x = ((RealFuncExtMult(A)).[a,f]).x proof let x be Element of A; thus h.x = a*(f.x) by A1 .= ((RealFuncExtMult(A)).[a,f]).x by Def4; end; hence h = (RealFuncExtMult(A)).[a,f] by FUNCT_2:113; end; hence thesis; end; reserve u,v,w for VECTOR of RLSStruct(#Funcs(A,REAL), (RealFuncZero(A)),RealFuncAdd(A),RealFuncExtMult(A)#); theorem Th16: (RealFuncAdd(A)).(f,g) = (RealFuncAdd(A)).(g,f) proof now let x be Element of A; thus ((RealFuncAdd(A)).(f,g)).x = g.x + f.x by Th10 .= ((RealFuncAdd(A)).(g,f)).x by Th10; end; hence thesis by FUNCT_2:113; end; theorem Th17: (RealFuncAdd(A)).(f,(RealFuncAdd(A)).(g,h)) = (RealFuncAdd(A)).((RealFuncAdd(A)).(f,g),h) proof now let x be Element of A; thus ((RealFuncAdd(A)).(f,(RealFuncAdd(A)).(g,h))).x = f.x + ((RealFuncAdd(A)).(g,h)).x by Th10 .= f.x + (g.x + h.x) by Th10 .= (f.x + g.x) + h.x by XCMPLX_1:1 .= ((RealFuncAdd(A)).(f,g)).x + h.x by Th10 .= ((RealFuncAdd(A)).((RealFuncAdd(A)).(f,g),h)).x by Th10; end; hence thesis by FUNCT_2:113; end; theorem Th18: (RealFuncMult(A)).(f,g) = (RealFuncMult(A)).(g,f) proof now let x be Element of A; thus ((RealFuncMult(A)).(f,g)).x = g.x * f.x by Th11 .= ((RealFuncMult(A)).(g,f)).x by Th11; end; hence thesis by FUNCT_2:113; end; theorem Th19: (RealFuncMult(A)).(f,(RealFuncMult(A)).(g,h)) = (RealFuncMult(A)).((RealFuncMult(A)).(f,g),h) proof now let x be Element of A; thus ((RealFuncMult(A)).(f,(RealFuncMult(A)).(g,h))).x = f.x * ((RealFuncMult(A)).(g,h)).x by Th11 .= f.x * (g.x * h.x) by Th11 .= (f.x * g.x) * h.x by XCMPLX_1:4 .= ((RealFuncMult(A)).(f,g)).x * h.x by Th11 .= ((RealFuncMult(A)).((RealFuncMult(A)).(f,g),h)).x by Th11; end; hence thesis by FUNCT_2:113; end; theorem Th20: (RealFuncMult(A)).(RealFuncUnit(A),f) = f proof now let x be Element of A; thus ((RealFuncMult(A)).(RealFuncUnit(A),f)).x= (RealFuncUnit(A)).x * f.x by Th11 .=1 * f.x by Th12 .= f.x; end; hence thesis by FUNCT_2:113; end; theorem Th21: (RealFuncAdd(A)).(RealFuncZero(A),f) = f proof now let x be Element of A; thus ((RealFuncAdd(A)).(RealFuncZero(A),f)).x = (RealFuncZero(A)).x + f.x by Th10 .= 0 + f.x by Th13 .= f.x; end; hence thesis by FUNCT_2:113; end; theorem Th22: (RealFuncAdd(A)).(f,(RealFuncExtMult(A)).[-1,f]) = RealFuncZero(A) proof now let x be Element of A; set y=f.x; thus ((RealFuncAdd(A)).(f,(RealFuncExtMult(A)).[-1,f])).x = f.x + ((RealFuncExtMult(A)).[-1,f]).x by Th10 .= f.x + ((-1)*y) by Th15 .= f.x + (-(1*y)) by XCMPLX_1:175 .= 0 by XCMPLX_0:def 6 .= (RealFuncZero(A)).x by Th13; end; hence thesis by FUNCT_2:113; end; theorem Th23: (RealFuncExtMult(A)).[1,f] = f proof now let x be Element of A; thus ((RealFuncExtMult(A)).[1 qua Real,f]).x = 1*(f.x) by Th15 .= f.x; end; hence thesis by FUNCT_2:113; end; theorem Th24: (RealFuncExtMult(A)).[a,(RealFuncExtMult(A)).[b,f]] = (RealFuncExtMult(A)).[a*b,f] proof now let x be Element of A; thus ((RealFuncExtMult(A)).[a,(RealFuncExtMult(A)).[b,f]]).x = a*(((RealFuncExtMult(A)).[b,f]).x) by Th15 .= a*(b*(f.x)) by Th15 .= (a*b)*(f.x) by XCMPLX_1:4 .= ((RealFuncExtMult(A)).[a*b,f]).x by Th15; end; hence thesis by FUNCT_2:113; end; theorem Th25: (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,f]) = (RealFuncExtMult(A)).[a+b,f] proof now let x be Element of A; thus ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,f])).x = ((RealFuncExtMult(A)).[a,f]).x + ((RealFuncExtMult(A)).[b,f]).x by Th10 .= a*(f.x) + ((RealFuncExtMult(A)).[b,f]).x by Th15 .= a*(f.x) + b*(f.x) by Th15 .= (a+b)*(f.x) by XCMPLX_1:8 .= ((RealFuncExtMult(A)).[a+b,f]).x by Th15; end; hence thesis by FUNCT_2:113; end; Lm2: (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[a,g]) = (RealFuncExtMult(A)).[a,(RealFuncAdd(A)).(f,g)] proof now let x be Element of A; thus ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[a,g])).x = ((RealFuncExtMult(A)).[a,f]).x + ((RealFuncExtMult(A)).[a,g]).x by Th10 .= a*(f.x) + ((RealFuncExtMult(A)).[a,g]).x by Th15 .= a*(f.x) + a*(g.x) by Th15 .= a*(f.x + g.x) by XCMPLX_1:8 .= a*(((RealFuncAdd(A)).(f,g)).x) by Th10 .= ((RealFuncExtMult(A)).[a,(RealFuncAdd(A)).(f,g)]).x by Th15; end; hence thesis by FUNCT_2:113; end; theorem Th26: (RealFuncMult(A)).(f,(RealFuncAdd(A)).(g,h)) = (RealFuncAdd(A)).((RealFuncMult(A)).(f,g),(RealFuncMult(A)).(f,h)) proof now let x be Element of A; thus ((RealFuncMult(A)).(f,(RealFuncAdd(A)).(g,h))).x = f.x * (((RealFuncAdd(A)).(g,h)).x) by Th11 .= f.x * (g.x + h.x) by Th10 .= (f.x * g.x) + (f.x * h.x) by XCMPLX_1:8 .= ((RealFuncMult(A)).(f,g)).x + (f.x * h.x) by Th11 .= ((RealFuncMult(A)).(f,g)).x + ((RealFuncMult(A)).(f,h)).x by Th11 .= ((RealFuncAdd(A)). ((RealFuncMult(A)).(f,g),(RealFuncMult(A)).(f,h))).x by Th10; end; hence thesis by FUNCT_2:113; end; theorem Th27: (RealFuncMult(A)).((RealFuncExtMult(A)).[a,f],g) = (RealFuncExtMult(A)).[a,(RealFuncMult(A)).(f,g)] proof now let x be Element of A; thus ((RealFuncMult(A)). ((RealFuncExtMult(A)).[a,f],g)).x = ((RealFuncExtMult(A)).[a,f]).x * g.x by Th11 .= (a*(f.x)) * g.x by Th15 .= a*(f.x * g.x) by XCMPLX_1:4 .= a*(((RealFuncMult(A)).(f,g)).x) by Th11 .= ((RealFuncExtMult(A)).[a,(RealFuncMult(A)).(f,g)]).x by Th15; end; hence thesis by FUNCT_2:113; end; theorem Th28: ex f,g st (for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z = 0)) & (for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1)) proof defpred P[set] means $1 = x1; deffunc F(set) = 0; deffunc G(set) = 1; A1: for z st z in A holds (P[z] implies G(z) in REAL) & (not P[z] implies F(z) in REAL); consider f being Function of A,REAL such that A2: for z st z in A holds (P[z] implies f.z = G(z)) & (not P[z] implies f.z = F(z)) from Lambda1C(A1); A3: for z st z in A holds (P[z] implies F(z) in REAL) & (not P[z] implies G(z) in REAL); consider g being Function of A,REAL such that A4: for z st z in A holds (P[z] implies g.z = F(z)) & (not P[z] implies g.z = G(z)) from Lambda1C(A3); reconsider f,g as Element of Funcs(A,REAL) by FUNCT_2:11; take f,g; thus thesis by A2,A4; end; theorem Th29: (x1 in A & x2 in A & x1<>x2) & (for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z = 0)) & (for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1)) implies (for a,b st (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]) = RealFuncZero(A) holds a=0 & b=0) proof assume that A1: x1 in A & x2 in A & x1<>x2 and A2: for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z = 0) and A3: for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1); A4: f.x1=1 & f.x2=0 & g.x1=0 & g.x2=1 by A1,A2,A3; reconsider x1,x2 as Element of A by A1; let a,b; assume A5: (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]) = RealFuncZero(A); then A6: 0 = ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x1 by Th13 .= (((RealFuncExtMult(A)).[a,f]).x1) + (((RealFuncExtMult(A)).[b,g]).x1) by Th10 .= a*(f.x1) + (((RealFuncExtMult(A)).[b,g]).x1) by Th15 .= a + b*0 by A4,Th15 .= a; 0 = ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x2 by A5,Th13 .= (((RealFuncExtMult(A)).[a,f]).x2) + (((RealFuncExtMult(A)).[b,g]).x2) by Th10 .= a*(f.x2) + (((RealFuncExtMult(A)).[b,g]).x2) by Th15 .=0 + b*1 by A4,Th15 .= b; hence a=0 & b=0 by A6; end; theorem x1 in A & x2 in A & x1<>x2 implies (ex f,g st for a,b st (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]) = RealFuncZero(A) holds a=0 & b=0) proof assume A1: x1 in A & x2 in A & x1<>x2; consider f,g such that A2: for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z = 0) and A3: for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1) by Th28; take f,g; let a,b; assume (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]) = RealFuncZero(A); hence thesis by A1,A2,A3,Th29; end; theorem Th31: A = {x1,x2} & x1<>x2 & (for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z = 0)) & (for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1)) implies for h holds (ex a,b st h = (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])) proof assume that A1: A = {x1,x2} & x1<>x2 and A2: for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z = 0) and A3: for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1); x1 in A & x2 in A by A1,TARSKI:def 2; then A4: f.x1=1 & f.x2=0 & g.x1=0 & g.x2=1 by A1,A2,A3; reconsider x1,x2 as Element of A by A1,TARSKI:def 2; let h; take a = h.x1 , b = h.x2; now let x be Element of A; A5: x = x1 or x = x2 by A1,TARSKI:def 2; A6: ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x1 = (((RealFuncExtMult(A)).[a,f]).x1) + (((RealFuncExtMult(A)).[b,g]).x1) by Th10 .= a*(f.x1) + (((RealFuncExtMult(A)).[b,g]).x1) by Th15 .= a + b*0 by A4,Th15 .= h.x1; ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x2 = (((RealFuncExtMult(A)).[a,f]).x2) + (((RealFuncExtMult(A)).[b,g]).x2) by Th10 .= a*(f.x2) + (((RealFuncExtMult(A)).[b,g]).x2) by Th15 .= 0 + b*1 by A4,Th15 .= h.x2; hence h.x = ((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x by A5,A6; end; hence thesis by FUNCT_2:113; end; theorem A = {x1,x2} & x1<>x2 implies ex f,g st (for h holds (ex a,b st h = (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]))) proof assume A1: A = {x1,x2} & x1<>x2; consider f,g such that A2: for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z =0) and A3: for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1) by Th28; take f,g; let h; thus thesis by A1,A2,A3,Th31; end; theorem Th33: (A = {x1,x2} & x1<>x2) implies (ex f,g st (for a,b st (RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]) = RealFuncZero(A) holds a=0 & b=0) & (for h holds (ex a,b st h = (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])))) proof assume A1: A = {x1,x2} & x1<>x2; then A2: x1 in A & x2 in A & x1<>x2 by TARSKI:def 2; consider f,g such that A3: for z st z in A holds (z=x1 implies f.z = 1) & (z<>x1 implies f.z =0) and A4: for z st z in A holds (z=x1 implies g.z = 0) & (z<>x1 implies g.z = 1) by Th28; take f,g; thus thesis by A1,A2,A3,A4,Th29,Th31; end; theorem Th34: RLSStruct(#Funcs(A,REAL), (RealFuncZero(A)),RealFuncAdd(A),RealFuncExtMult(A)#) is RealLinearSpace proof set IT = RLSStruct(#Funcs(A,REAL), (RealFuncZero(A)),RealFuncAdd(A),RealFuncExtMult(A)#); IT is Abelian add-associative right_zeroed right_complementable RealLinearSpace-like proof thus u+v = v+u proof reconsider v' = v, u' = u as Element of Funcs(A,REAL); thus u+v = (RealFuncAdd(A)).[u',v'] by RLVECT_1:def 3 .= (RealFuncAdd(A)).(u',v') by BINOP_1:def 1 .= (RealFuncAdd(A)).(v',u') by Th16 .= (RealFuncAdd(A)).[v',u'] by BINOP_1:def 1 .= v+u by RLVECT_1:def 3; end; thus (u+v)+w = u+(v+w) proof reconsider v'=v, u'=u, w'=w as Element of Funcs(A,REAL); thus (u+v)+w = (RealFuncAdd(A)).[u+v,w'] by RLVECT_1:def 3 .= (RealFuncAdd(A)).(u+v,w') by BINOP_1:def 1 .= (RealFuncAdd(A)).((RealFuncAdd(A)).[u',v'],w') by RLVECT_1:def 3 .= (RealFuncAdd(A)).((RealFuncAdd(A)).(u',v'),w') by BINOP_1:def 1 .= (RealFuncAdd(A)).(u',(RealFuncAdd(A)).(v',w')) by Th17 .= (RealFuncAdd(A)).(u',(RealFuncAdd(A)).[v',w']) by BINOP_1:def 1 .= (RealFuncAdd(A)).[u',(RealFuncAdd(A)).[v',w']] by BINOP_1:def 1 .= (RealFuncAdd(A)).[u',v+w] by RLVECT_1:def 3 .= u+(v+w) by RLVECT_1:def 3; end; thus u+0.IT = u proof reconsider u'=u as Element of Funcs(A,REAL); thus u+0.IT = (RealFuncAdd(A)).[u',0.IT] by RLVECT_1:def 3 .= (RealFuncAdd(A)).[u',RealFuncZero(A)] by RLVECT_1:def 2 .= (RealFuncAdd(A)).(u',RealFuncZero(A)) by BINOP_1:def 1 .= (RealFuncAdd(A)).(RealFuncZero(A),u') by Th16 .= u by Th21; end; thus ex w st u+w = 0.IT proof reconsider u' = u as Element of Funcs(A,REAL); reconsider w = (RealFuncExtMult(A)).[-1,u'] as VECTOR of IT; take w; thus u+w = (RealFuncAdd(A)).[u',(RealFuncExtMult(A)).[-1,u']] by RLVECT_1:def 3 .= (RealFuncAdd(A)).(u',(RealFuncExtMult(A)).[-1,u']) by BINOP_1:def 1 .= RealFuncZero(A) by Th22 .= 0.IT by RLVECT_1:def 2; end; thus a*(u+v) = a*u + a *v proof reconsider v' = v, u' = u as Element of Funcs(A,REAL); reconsider w = (RealFuncExtMult(A)).[a,u'],w' = (RealFuncExtMult(A)).[a,v'] as VECTOR of IT; thus a*(u+v) = (RealFuncExtMult(A)).[a,u+v] by RLVECT_1:def 4 .=(RealFuncExtMult(A)).[a,(RealFuncAdd(A)).[u',v']] by RLVECT_1:def 3 .= (RealFuncExtMult(A)).[a,(RealFuncAdd(A)).(u',v')] by BINOP_1:def 1 .=(RealFuncAdd(A)).(w,w') by Lm2 .= (RealFuncAdd(A)).[w,w'] by BINOP_1:def 1 .= w + w' by RLVECT_1:def 3 .= w + (a*v) by RLVECT_1:def 4 .= (a*u) + (a*v) by RLVECT_1:def 4; end; thus (a+b)*v = a*v + b*v proof reconsider v' = v as Element of Funcs(A,REAL); reconsider w = (RealFuncExtMult(A)).[a,v'],w' = (RealFuncExtMult(A)).[b,v'] as VECTOR of IT; thus (a+b)*v = (RealFuncExtMult(A)).[a+b,v'] by RLVECT_1:def 4 .= (RealFuncAdd(A)).(w,w') by Th25 .= (RealFuncAdd(A)).[w,w'] by BINOP_1:def 1 .= w + w' by RLVECT_1:def 3 .= w + (b*v) by RLVECT_1:def 4 .= (a*v) + (b*v) by RLVECT_1:def 4; end; thus (a*b)*v = a*(b*v) proof reconsider v'=v as Element of Funcs(A,REAL); thus (a*b)*v = (RealFuncExtMult(A)).[a*b,v'] by RLVECT_1:def 4 .= (RealFuncExtMult(A)).[a,(RealFuncExtMult(A)).[b,v']] by Th24 .= (RealFuncExtMult(A)).[a,b*v] by RLVECT_1:def 4 .= a*(b*v) by RLVECT_1:def 4; end; thus 1*v = v proof reconsider v'=v as Element of Funcs(A,REAL); thus 1*v = (RealFuncExtMult(A)).[1,v'] by RLVECT_1:def 4 .= v by Th23; end; end; hence thesis; end; definition let A; func RealVectSpace(A) -> strict RealLinearSpace equals :Def7: RLSStruct(#Funcs(A,REAL), (RealFuncZero(A)),RealFuncAdd(A),RealFuncExtMult(A)#); coherence by Th34; end; Lm3: ex A,x1,x2 st A={x1,x2} & x1<>x2 proof reconsider A={1,2} as non empty set; take A; thus thesis; end; canceled 2; theorem ex V being strict RealLinearSpace st (ex u,v being VECTOR of V st (for a,b st a*u + b*v = 0.V holds a=0 & b=0) & (for w being VECTOR of V ex a,b st w = a*u + b*v)) proof consider A,x1,x2 such that A1: A={x1,x2} & x1<>x2 by Lm3; take V = RealVectSpace(A); consider f,g such that A2: (for a,b st (RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]) = RealFuncZero(A) holds a=0 & b=0) & (for h holds (ex a,b st h = (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]))) by A1,Th33; A3: V=RLSStruct(#Funcs(A,REAL), (RealFuncZero(A)),RealFuncAdd(A),RealFuncExtMult(A)#) by Def7; then reconsider u=f, v=g as VECTOR of V; take u,v; thus for a,b st a*u + b*v = 0.V holds a=0 & b=0 proof let r,s be Real such that A4: r*u + s*v=0.V; A5: r*u = (RealFuncExtMult(A)).[r,f] & s*v = (RealFuncExtMult(A)).[s,g] by A3,RLVECT_1:def 4; reconsider u' = r*u,v' = s*v as Element of Funcs(A,REAL) by A3; 0.V = (RealFuncAdd(A)).[u', v'] by A3,A4,RLVECT_1:def 3 .= (RealFuncAdd(A)).((RealFuncExtMult(A)).[r,f], (RealFuncExtMult(A)).[s,g]) by A5,BINOP_1:def 1; then (RealFuncAdd(A)).((RealFuncExtMult(A)).[r,f], (RealFuncExtMult(A)).[s,g]) = RealFuncZero(A) by A3,RLVECT_1:def 2; hence r=0 & s=0 by A2; end; thus for w being VECTOR of V ex a,b st w = a*u + b*v proof let w be VECTOR of V; reconsider h=w as Element of Funcs(A,REAL) by A3; consider a,b such that A6: h = (RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]) by A2; h = (RealFuncAdd(A)). (a*u,(RealFuncExtMult(A)).[b,g]) by A3,A6,RLVECT_1:def 4 .= (RealFuncAdd(A)).(a*u,b*v) by A3,RLVECT_1:def 4 .= (RealFuncAdd(A)).[a*u,b*v] by BINOP_1:def 1 .= a*u + b*v by A3,RLVECT_1:def 3; hence thesis; end; end; definition let A; canceled 4; func RRing(A) -> strict doubleLoopStr equals :Def12: doubleLoopStr(#Funcs(A,REAL),RealFuncAdd(A),RealFuncMult(A), (RealFuncUnit(A)),(RealFuncZero(A))#); correctness; end; definition let A; cluster RRing A -> non empty; coherence proof RRing A = doubleLoopStr(#Funcs(A,REAL),RealFuncAdd(A),RealFuncMult(A), (RealFuncUnit(A)),(RealFuncZero(A))#) by Def12; hence the carrier of RRing A is non empty; end; end; canceled 4; theorem Th42: for x,y,z being Element of RRing(A) holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.RRing(A)) = x & (ex t being Element of RRing(A) st x+t=(0.RRing(A))) & x*y = y*x & (x*y)*z = x*(y*z) & x*(1_ RRing(A)) = x & (1_ RRing(A))*x = x & x*(y+z) = x*y + x*z & (y+z)*x = y*x + z*x proof let x,y,z be Element of RRing(A); A1: RRing(A)=doubleLoopStr(#Funcs(A,REAL),RealFuncAdd(A),RealFuncMult(A), (RealFuncUnit(A)),(RealFuncZero(A))#) by Def12; A2:now let x,y be Element of RRing(A); reconsider f=x, g=y as Element of Funcs(A,REAL) by A1; thus x*y = (RealFuncMult(A)).(f,g) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(g,f) by Th18 .= y*x by A1,VECTSP_1:def 10; end; set IT = RRing(A); reconsider f=x, g=y, h=z as Element of Funcs(A,REAL) by A1; thus x+y = (RealFuncAdd(A)).(f,g) by A1,RLVECT_1:5 .= (the add of IT).(g,f) by A1,Th16 .= y+x by RLVECT_1:5; thus (x+y)+z = (the add of IT).(x+y,z) by RLVECT_1:5 .= (RealFuncAdd(A)).((RealFuncAdd(A)).(f,g),h) by A1,RLVECT_1:5 .= (RealFuncAdd(A)).(f,(the add of IT).(y,z)) by A1,Th17 .= (RealFuncAdd(A)).(f,y+z) by RLVECT_1:5 .= x+(y+z) by A1,RLVECT_1:5; thus x+(0.RRing(A)) = (RealFuncAdd(A)).(f,0.IT) by A1,RLVECT_1:5 .= (RealFuncAdd(A)).(f,RealFuncZero(A)) by A1,RLVECT_1:def 2 .= (RealFuncAdd(A)).(RealFuncZero(A),f) by Th16 .= x by Th21; thus ex t being Element of RRing(A) st x+t=(0.RRing(A)) proof set h = (RealFuncExtMult(A)).[-1,f]; reconsider t=h as Element of IT by A1; take t; thus x+t = (RealFuncAdd(A)).(f,h) by A1,RLVECT_1:5 .= RealFuncZero(A) by Th22 .= 0.IT by A1,RLVECT_1:def 2; end; thus x*y = y*x by A2; thus (x*y)*z = (RealFuncMult(A)).(x*y,h) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).((RealFuncMult(A)).(f,g),h) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(f,(RealFuncMult(A)).(g,h)) by Th19 .= (RealFuncMult(A)).(f,y*z) by A1,VECTSP_1:def 10 .= x*(y*z) by A1,VECTSP_1:def 10; thus x*(1_ RRing(A)) = (RealFuncMult(A)).(f,1_ IT) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(f,RealFuncUnit(A)) by A1,VECTSP_1:def 9 .= (RealFuncMult(A)).(RealFuncUnit(A),f) by Th18 .= x by Th20; hence (1_ RRing(A))*x = x by A2; thus x*(y+z) = (RealFuncMult(A)).(f,y+z) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(f,(RealFuncAdd(A)).(g,h)) by A1,RLVECT_1:5 .= (RealFuncAdd(A)). ((RealFuncMult(A)).(f,g),(RealFuncMult(A)).(f,h)) by Th26 .= (RealFuncAdd(A)).(x*y,(RealFuncMult(A)).(f,h)) by A1,VECTSP_1:def 10 .= (RealFuncAdd(A)).(x*y,x*z) by A1,VECTSP_1:def 10 .= x*y + x*z by A1,RLVECT_1:5; hence (y+z)*x = x*y + x*z by A2 .= y*x + x*z by A2 .= y*x + z*x by A2; end; definition cluster strict Abelian add-associative right_zeroed right_complementable associative commutative right_unital right-distributive (non empty doubleLoopStr); existence proof consider A; for x,y,z being Element of RRing(A) holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.RRing(A)) = x & (ex t being Element of RRing(A) st x+t=(0.RRing(A))) & x*y = y*x & (x*y)*z = x*(y*z) & x*(1_ RRing(A)) = x & x*(y+z) = x*y + x*z by Th42; then RRing(A) is Abelian add-associative right_zeroed right_complementable associative commutative right_unital right-distributive by RLVECT_1:def 5,def 6,def 7,def 8,VECTSP_1:def 11,def 13,def 16,def 17; hence thesis; end; end; definition mode Ring is Abelian add-associative right_zeroed right_complementable associative left_unital right_unital distributive (non empty doubleLoopStr); end; theorem RRing(A) is commutative Ring proof for x,y,z being Element of RRing(A) holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.RRing(A)) = x & (ex t being Element of RRing(A) st x+t=(0.RRing(A))) & x*y = y*x & (x*y)*z = x*(y*z) & x*(1_ RRing(A)) = x & (1_ RRing(A))*x = x & x*(y+z) = x*y + x*z & (y+z)*x = y*x + z*x by Th42; hence thesis by RLVECT_1:def 5,def 6,def 7,def 8,VECTSP_1:def 13,def 16,def 17,def 18,def 19; end; definition struct(doubleLoopStr,RLSStruct) AlgebraStr (# carrier -> set, mult,add -> (BinOp of the carrier), Mult -> (Function of [:REAL,the carrier:],the carrier), unity,Zero -> Element of the carrier #); end; definition cluster non empty AlgebraStr; existence proof consider X being non empty set, m,a being BinOp of X, M being Function of [:REAL,X:],X, u,Z being Element of X; take AlgebraStr(#X,m,a,M,u,Z#); thus the carrier of AlgebraStr(#X,m,a,M,u,Z#) is non empty; end; end; definition let A; canceled 6; func RAlgebra(A) -> strict AlgebraStr equals :Def19: AlgebraStr(#Funcs(A,REAL),RealFuncMult(A),RealFuncAdd(A), RealFuncExtMult(A),(RealFuncUnit(A)),(RealFuncZero(A))#); correctness; end; definition let A; cluster RAlgebra(A) -> non empty; coherence proof RAlgebra A = AlgebraStr(#Funcs(A,REAL),RealFuncMult(A),RealFuncAdd(A), RealFuncExtMult(A),(RealFuncUnit(A)),(RealFuncZero(A))#) by Def19; hence the carrier of RAlgebra A is non empty; end; end; canceled 5; theorem Th49: for x,y,z being Element of RAlgebra(A) for a,b holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.RAlgebra(A)) = x & (ex t being Element of RAlgebra(A) st x+t=(0.RAlgebra(A))) & x*y = y*x & (x*y)*z = x*(y*z) & x*(1_ RAlgebra(A)) = x & x*(y+z) = x*y + x*z & a*(x*y) = (a*x)*y & a*(x+y) = a*x + a*y & (a+b)*x = a*x + b*x & (a*b)*x = a*(b*x) proof let x,y,z be Element of RAlgebra(A); let a,b; A1: RAlgebra(A)= AlgebraStr(#Funcs(A,REAL),RealFuncMult(A),RealFuncAdd(A), RealFuncExtMult(A),(RealFuncUnit(A)),(RealFuncZero(A))#) by Def19; set IT = RAlgebra(A); reconsider f=x, g=y, h=z as Element of Funcs(A,REAL) by A1; thus x+y = (RealFuncAdd(A)).(f,g) by A1,RLVECT_1:5 .= (the add of IT).(g,f) by A1,Th16 .= y+x by RLVECT_1:5; thus (x+y)+z = (the add of IT).(x+y,z) by RLVECT_1:5 .= (RealFuncAdd(A)).((RealFuncAdd(A)).(f,g),h) by A1,RLVECT_1:5 .= (RealFuncAdd(A)).(f,(the add of IT).(y,z)) by A1,Th17 .= (RealFuncAdd(A)).(f,y+z) by RLVECT_1:5 .= x+(y+z) by A1,RLVECT_1:5; thus x+(0.RAlgebra(A)) = (RealFuncAdd(A)).(f,0.IT) by A1,RLVECT_1:5 .= (RealFuncAdd(A)).(f,RealFuncZero(A)) by A1,RLVECT_1:def 2 .= (RealFuncAdd(A)).(RealFuncZero(A),f) by Th16 .= x by Th21; thus ex t being Element of RAlgebra(A) st x+t=(0.RAlgebra(A)) proof set h = (RealFuncExtMult(A)).[-1,f]; reconsider t=h as Element of IT by A1; take t; thus x+t = (RealFuncAdd(A)).(f,h) by A1,RLVECT_1:5 .= RealFuncZero(A) by Th22 .= 0.IT by A1,RLVECT_1:def 2; end; thus x*y = (RealFuncMult(A)).(f,g) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(g,f) by Th18 .= y*x by A1,VECTSP_1:def 10; thus (x*y)*z = (RealFuncMult(A)).(x*y,h) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).((RealFuncMult(A)).(f,g),h) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(f,(RealFuncMult(A)).(g,h)) by Th19 .= (RealFuncMult(A)).(f,y*z) by A1,VECTSP_1:def 10 .= x*(y*z) by A1,VECTSP_1:def 10; thus x*(1_ RAlgebra(A)) = (RealFuncMult(A)).(f,1_ IT) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(f,RealFuncUnit(A)) by A1,VECTSP_1:def 9 .= (RealFuncMult(A)).(RealFuncUnit(A),f) by Th18 .= x by Th20; thus x*(y+z) = (RealFuncMult(A)).(f,y+z) by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).(f,(RealFuncAdd(A)).(g,h)) by A1,RLVECT_1:5 .= (RealFuncAdd(A)). ((RealFuncMult(A)).(f,g),(RealFuncMult(A)).(f,h)) by Th26 .= (RealFuncAdd(A)).(x*y,(RealFuncMult(A)).(f,h)) by A1,VECTSP_1:def 10 .= (RealFuncAdd(A)).(x*y,x*z) by A1,VECTSP_1:def 10 .= x*y + x*z by A1,RLVECT_1:5; thus a*(x*y) = (RealFuncExtMult(A)).[a,x*y] by A1,RLVECT_1:def 4 .= (RealFuncExtMult(A)).[a,(RealFuncMult(A)).(f,g)] by A1,VECTSP_1:def 10 .= (RealFuncMult(A)).((RealFuncExtMult(A)).[a,f],g) by Th27 .= (RealFuncMult(A)).(a*x,g) by A1,RLVECT_1:def 4 .= (a*x)*y by A1,VECTSP_1:def 10; thus a*(x+y) = (RealFuncExtMult(A)).[a,x+y] by A1,RLVECT_1:def 4 .= (RealFuncExtMult(A)).[a,(RealFuncAdd(A)).(f,g)] by A1,RLVECT_1:5 .= (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[a,g]) by Lm2 .= (RealFuncAdd(A)).(a*x,(RealFuncExtMult(A)).[a,g]) by A1,RLVECT_1:def 4 .= (RealFuncAdd(A)).(a*x,a*y) by A1,RLVECT_1:def 4 .= (a*x) + (a*y) by A1,RLVECT_1:5; thus (a+b)*x = (RealFuncExtMult(A)).[a+b,f] by A1,RLVECT_1:def 4 .= (RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,f]) by Th25 .= (RealFuncAdd(A)).(a*x,(RealFuncExtMult(A)).[b,f]) by A1,RLVECT_1:def 4 .= (RealFuncAdd(A)).(a*x,b*x) by A1,RLVECT_1:def 4 .= (a*x) + (b*x) by A1,RLVECT_1:5; thus (a*b)*x = (RealFuncExtMult(A)).[a*b,f] by A1,RLVECT_1:def 4 .= (RealFuncExtMult(A)). [a,(RealFuncExtMult(A)).[b,f]] by Th24 .= (RealFuncExtMult(A)).[a,b*x] by A1,RLVECT_1:def 4 .= a*(b*x) by A1,RLVECT_1:def 4; end; definition let IT be non empty AlgebraStr; attr IT is Algebra-like means :Def20: for x,y,z being Element of IT for a,b holds x*(1_ IT) = x & x*(y+z) = x*y + x*z & a*(x*y) = (a*x)*y & a*(x+y) = a*x + a*y & (a+b)*x = a*x + b*x & (a*b)*x = a*(b*x); end; definition cluster strict Abelian add-associative right_zeroed right_complementable commutative associative Algebra-like (non empty AlgebraStr); existence proof consider A; for x,y,z being Element of RAlgebra(A) for a,b holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.RAlgebra(A)) = x & (ex t being Element of RAlgebra(A) st x+t=(0.RAlgebra(A))) & x*y = y*x & (x*y)*z = x*(y*z) & x*(1_ RAlgebra(A)) = x & x*(y+z) = x*y + x*z & a*(x*y) = (a*x)*y & a*(x+y) = a*x + a*y & (a+b)*x = a*x + b*x & (a*b)*x = a*(b*x) by Th49; then RAlgebra(A) is Abelian add-associative right_zeroed right_complementable Algebra-like commutative associative by Def20,RLVECT_1:def 5,def 6,def 7,def 8,VECTSP_1:def 16,def 17; hence thesis; end; end; definition mode Algebra is Abelian add-associative right_zeroed right_complementable commutative associative Algebra-like (non empty AlgebraStr); end; theorem RAlgebra(A) is Algebra proof for x,y,z being Element of RAlgebra(A) for a,b holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.RAlgebra(A)) = x & (ex t being Element of RAlgebra(A) st x+t=(0.RAlgebra(A))) & x*y = y*x & (x*y)*z = x*(y*z) & x*(1_ RAlgebra(A)) = x & x*(y+z) = x*y + x*z & a*(x*y) = (a*x)*y & a*(x+y) = a*x + a*y & (a+b)*x = a*x + b*x & (a*b)*x = a*(b*x) by Th49; hence thesis by Def20,RLVECT_1:def 5,def 6,def 7,def 8,VECTSP_1:def 16,def 17; end;

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