let L be Z_Lattice; :: thesis: for f being Function of L,INT.Ring
for F, G being FinSequence of L
for v being Vector of L holds ScFS (v,f,(F ^ G)) = (ScFS (v,f,F)) ^ (ScFS (v,f,G))
let f be Function of L,INT.Ring; :: thesis: for F, G being FinSequence of L
for v being Vector of L holds ScFS (v,f,(F ^ G)) = (ScFS (v,f,F)) ^ (ScFS (v,f,G))
let F, G be FinSequence of L; :: thesis: for v being Vector of L holds ScFS (v,f,(F ^ G)) = (ScFS (v,f,F)) ^ (ScFS (v,f,G))
let v be Vector of L; :: thesis: ScFS (v,f,(F ^ G)) = (ScFS (v,f,F)) ^ (ScFS (v,f,G))
set H = (ScFS (v,f,F)) ^ (ScFS (v,f,G));
set I = F ^ G;
A1: len F = len (ScFS (v,f,F)) by defScFS;
A2: len ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) = (len (ScFS (v,f,F))) + (len (ScFS (v,f,G))) by FINSEQ_1:22
.= (len F) + (len (ScFS (v,f,G))) by defScFS
.= (len F) + (len G) by defScFS
.= len (F ^ G) by FINSEQ_1:22 ;
A3: len G = len (ScFS (v,f,G)) by defScFS;

now :: thesis: for k being Nat st k in dom ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) holds
((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . k = <;v,((f . ((F ^ G) /. k)) * ((F ^ G) /. k));>

let k be Nat; :: thesis: ( k in dom ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) implies ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . b₁ = <;v,((f . ((F ^ G) /. b₁)) * ((F ^ G) /. b₁));> )
assume A4: k in dom ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) ; :: thesis: ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . b₁ = <;v,((f . ((F ^ G) /. b₁)) * ((F ^ G) /. b₁));>

per cases ( k in dom (ScFS (v,f,F)) or ex n being Nat st
( n in dom (ScFS (v,f,G)) & k = (len (ScFS (v,f,F))) + n ) ) by A4, FINSEQ_1:25;

suppose A5: k in dom (ScFS (v,f,F)) ; :: thesis: ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . b₁ = <;v,((f . ((F ^ G) /. b₁)) * ((F ^ G) /. b₁));>

then A6: k in dom F by A1, FINSEQ_3:29;
then A7: k in dom (F ^ G) by FINSEQ_3:22;
A8: F /. k = F . k by A6, PARTFUN1:def 6
.= (F ^ G) . k by A6, FINSEQ_1:def 7
.= (F ^ G) /. k by A7, PARTFUN1:def 6 ;
thus ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . k = (ScFS (v,f,F)) . k by A5, FINSEQ_1:def 7
.= <;v,((f . ((F ^ G) /. k)) * ((F ^ G) /. k));> by A5, A8, defScFS ; :: thesis: verum

end;

suppose A9: ex n being Nat st
( n in dom (ScFS (v,f,G)) & k = (len (ScFS (v,f,F))) + n ) ; :: thesis: ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . b₁ = <;v,((f . ((F ^ G) /. b₁)) * ((F ^ G) /. b₁));>

A10: k in dom (F ^ G) by A2, A4, FINSEQ_3:29;
consider n being Nat such that
A11: n in dom (ScFS (v,f,G)) and
A12: k = (len (ScFS (v,f,F))) + n by A9;
A13: n in dom G by A3, A11, FINSEQ_3:29;
then A14: G /. n = G . n by PARTFUN1:def 6
.= (F ^ G) . k by A1, A12, A13, FINSEQ_1:def 7
.= (F ^ G) /. k by A10, PARTFUN1:def 6 ;
thus ((ScFS (v,f,F)) ^ (ScFS (v,f,G))) . k = (ScFS (v,f,G)) . n by A11, A12, FINSEQ_1:def 7
.= <;v,((f . ((F ^ G) /. k)) * ((F ^ G) /. k));> by A11, A14, defScFS ; :: thesis: verum

end;

end;

end;

hence ScFS (v,f,(F ^ G)) = (ScFS (v,f,F)) ^ (ScFS (v,f,G)) by A2, defScFS; :: thesis: verum