let n be Nat; :: thesis: for X being set
for f being PartFunc of REAL,REAL st X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} holds
Sgm (f .: X) = f * (Sgm X)
let X be set ; :: thesis: for f being PartFunc of REAL,REAL st X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} holds
Sgm (f .: X) = f * (Sgm X)
let f be PartFunc of REAL,REAL; :: thesis: ( X c= Seg n & X c= dom f & f | X is increasing & f .: X c= NAT \ {0} implies Sgm (f .: X) = f * (Sgm X) )
assume A1: X c= Seg n ; :: thesis: ( not X c= dom f or not f | X is increasing or not f .: X c= NAT \ {0} or Sgm (f .: X) = f * (Sgm X) )
then a1: X is included_in_Seg ;
then A2: rng (Sgm X) = X by FINSEQ_1:def 14;
assume A3: X c= dom f ; :: thesis: ( not f | X is increasing or not f .: X c= NAT \ {0} or Sgm (f .: X) = f * (Sgm X) )
assume A4: f | X is increasing ; :: thesis: ( not f .: X c= NAT \ {0} or Sgm (f .: X) = f * (Sgm X) )
assume A5: f .: X c= NAT \ {0} ; :: thesis: Sgm (f .: X) = f * (Sgm X)

per cases ( X misses dom f or X meets dom f ) ;

suppose A6: X misses dom f ; :: thesis: Sgm (f .: X) = f * (Sgm X)

then f .: X = {} by RELAT_1:118;
then Sgm (f .: X) = {} by FINSEQ_1:51;
hence Sgm (f .: X) = f * (Sgm X) by A2, A6, RELAT_1:44; :: thesis: verum

end;

suppose A8: X meets dom f ; :: thesis: Sgm (f .: X) = f * (Sgm X)

reconsider X9 = X as finite set by A1;
set fX = f .: X;
reconsider f9 = f as Function ;
AA: f9 .: X9 is finite ;
f .: X c= NAT \ {0} by A5;
then reconsider fX = f .: X as non empty finite natural-membered set by A8, AA, RELAT_1:118;
reconsider k = max fX as Nat by TARSKI:1;
set fs = f * (Sgm X);
rng (Sgm X) c= dom f by a1, A3, FINSEQ_1:def 14;
then reconsider fs = f * (Sgm X) as FinSequence by FINSEQ_1:16;
f .: (rng (Sgm X)) c= NAT \ {0} by a1, A5, FINSEQ_1:def 14;
then A9: rng fs c= NAT \ {0} by RELAT_1:127;
rng fs c= NAT by A9, XBOOLE_1:1;
then reconsider fs = fs as FinSequence of NAT by FINSEQ_1:def 4;

now :: thesis: for x being object st x in f .: X holds
x in Seg k

let x be object ; :: thesis: ( x in f .: X implies x in Seg k )
assume A10: x in f .: X ; :: thesis: x in Seg k
then reconsider k9 = x as Nat by A5;
not k9 in {0} by A5, A10, XBOOLE_0:def 5;
then k9 <> 0 by TARSKI:def 1;
then 0 + 1 < k9 + 1 by XREAL_1:6;
then A11: 1 <= k9 by NAT_1:13;
k9 <= k by A10, XXREAL_2:def 8;
hence x in Seg k by A11; :: thesis: verum

end;

then f .: X c= Seg k ;
then a12: f .: X is included_in_Seg ;

A13: now :: thesis: for l, m being Nat st 1 <= l & l < m & m <= len fs holds
fs . l < fs . m

A14: dom fs = Seg (len fs) by FINSEQ_1:def 3;
let l, m be Nat; :: thesis: ( 1 <= l & l < m & m <= len fs implies fs . l < fs . m )
assume that
A15: 1 <= l and
A16: l < m and
A17: m <= len fs ; :: thesis: fs . l < fs . m
set k19 = (Sgm X) . l;
l <= len fs by A16, A17, XXREAL_0:2;
then A18: l in dom fs by A15, A14;
then l in dom (Sgm X) by FUNCT_1:11;
then A19: (Sgm X) . l in X by A2, FUNCT_1:3;
set k29 = (Sgm X) . m;
1 <= m by A15, A16, XXREAL_0:2;
then A20: m in dom fs by A17, A14;
then A21: m in dom (Sgm X) by FUNCT_1:11;
then A22: (Sgm X) . m in X by A2, FUNCT_1:3;
reconsider k19 = (Sgm X) . l, k29 = (Sgm X) . m as Nat ;
m in Seg (len (Sgm X)) by A21, FINSEQ_1:def 3;
then m <= len (Sgm X) by FINSEQ_1:1;
then A23: k19 < k29 by a1, A15, A16, FINSEQ_1:def 14;
reconsider k19 = k19, k29 = k29 as Element of NAT by ORDINAL1:def 12;
reconsider k19 = k19, k29 = k29 as Element of REAL by XREAL_0:def 1;
(Sgm X) . l in dom f by A18, FUNCT_1:11;
then A24: k19 in X /\ (dom f) by A19, XBOOLE_0:def 4;
set k1 = fs . l;
set k2 = fs . m;
A27: fs . m = f . ((Sgm X) . m) by A20, FUNCT_1:12;
(Sgm X) . m in dom f by A20, FUNCT_1:11;
then A28: k29 in X /\ (dom f) by A22, XBOOLE_0:def 4;
fs . l = f . ((Sgm X) . l) by A18, FUNCT_1:12;
hence fs . l < fs . m by A4, A27, A23, A24, A28, RFUNCT_2:20; :: thesis: verum

end;

rng fs = f .: X by A2, RELAT_1:127;
hence Sgm (f .: X) = f * (Sgm X) by a12, A13, FINSEQ_1:def 14; :: thesis: verum

end;

end;