let f be FinSeq-Location ; :: thesis: for i being Instruction of SCM+FSA
for s being State of SCM+FSA st not f in UsedInt*Loc i holds
(Exec (i,s)) . f = s . f
let i be Instruction of SCM+FSA; :: thesis: for s being State of SCM+FSA st not f in UsedInt*Loc i holds
(Exec (i,s)) . f = s . f
let s be State of SCM+FSA; :: thesis: ( not f in UsedInt*Loc i implies (Exec (i,s)) . f = s . f )
assume A1: not f in UsedInt*Loc i ; :: thesis: (Exec (i,s)) . f = s . f
A4: InsCode i <= 12 by SCMFSA_2:16;

per cases ( InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 3 or InsCode i = 4 or InsCode i = 5 or InsCode i = 6 or InsCode i = 7 or InsCode i = 8 or InsCode i = 9 or InsCode i = 10 or InsCode i = 11 or InsCode i = 12 ) by A4, NAT_1:36;

suppose InsCode i = 0 ; :: thesis: (Exec (i,s)) . f = s . f

then i = halt SCM+FSA by SCMFSA_2:95;
hence (Exec (i,s)) . f = s . f by EXTPRO_1:def 3; :: thesis: verum

end;

suppose InsCode i = 1 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a, b being Int-Location st i = a := b by SCMFSA_2:30;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:63; :: thesis: verum

end;

suppose InsCode i = 2 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a, b being Int-Location st i = AddTo (a,b) by SCMFSA_2:31;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:64; :: thesis: verum

end;

suppose InsCode i = 3 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a, b being Int-Location st i = SubFrom (a,b) by SCMFSA_2:32;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:65; :: thesis: verum

end;

suppose InsCode i = 4 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a, b being Int-Location st i = MultBy (a,b) by SCMFSA_2:33;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:66; :: thesis: verum

end;

suppose InsCode i = 5 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a, b being Int-Location st i = Divide (a,b) by SCMFSA_2:34;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:67; :: thesis: verum

end;

suppose InsCode i = 6 ; :: thesis: (Exec (i,s)) . f = s . f

then ex l being Element of NAT st i = goto l by SCMFSA_2:35;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:69; :: thesis: verum

end;

suppose InsCode i = 7 ; :: thesis: (Exec (i,s)) . f = s . f

then ex l being Element of NAT ex a being Int-Location st i = a =0_goto l by SCMFSA_2:36;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:70; :: thesis: verum

end;

suppose InsCode i = 8 ; :: thesis: (Exec (i,s)) . f = s . f

then ex l being Element of NAT ex a being Int-Location st i = a >0_goto l by SCMFSA_2:37;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:71; :: thesis: verum

end;

suppose InsCode i = 9 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a, b being Int-Location ex g being FinSeq-Location st i = b := (g,a) by SCMFSA_2:38;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:72; :: thesis: verum

end;

suppose InsCode i = 10 ; :: thesis: (Exec (i,s)) . f = s . f

then consider a, b being Int-Location , g being FinSeq-Location such that
A5: i = (g,a) := b by SCMFSA_2:39;
UsedInt*Loc i = {g} by A5, Th37;
then f <> g by A1, TARSKI:def 1;
hence (Exec (i,s)) . f = s . f by A5, SCMFSA_2:73; :: thesis: verum

end;

suppose InsCode i = 11 ; :: thesis: (Exec (i,s)) . f = s . f

then ex a being Int-Location ex g being FinSeq-Location st i = a :=len g by SCMFSA_2:40;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:74; :: thesis: verum

end;

suppose InsCode i = 12 ; :: thesis: (Exec (i,s)) . f = s . f

then consider a being Int-Location , g being FinSeq-Location such that
A6: i = g :=<0,...,0> a by SCMFSA_2:41;
UsedInt*Loc i = {g} by A6, Th38;
then f <> g by A1, TARSKI:def 1;
hence (Exec (i,s)) . f = s . f by A6, SCMFSA_2:75; :: thesis: verum

end;

end;