let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being really-closed InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= P1 & I c= P2 holds
( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
let s be State of SCM+FSA; :: thesis: for I being really-closed InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= P1 & I c= P2 holds
( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
let I be really-closed InitHalting Program of SCM+FSA; :: thesis: ( Initialize ((intloc 0) .--> 1) c= s & I c= P1 & I c= P2 implies ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) ) )
assume that
A1: Initialize ((intloc 0) .--> 1) c= s and
A2: I c= P1 and
A3: I c= P2 ; :: thesis: ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
A4: P2 halts_on s by A1, A3, Def1;
A5: P1 halts_on s by A1, A2, Def1;

A6: now :: thesis: for l being Nat st CurInstr (P2,(Comput (P2,s,l))) = halt SCM+FSA holds
LifeSpan (P1,s) <= l

let l be Nat; :: thesis: ( CurInstr (P2,(Comput (P2,s,l))) = halt SCM+FSA implies LifeSpan (P1,s) <= l )
assume A7: CurInstr (P2,(Comput (P2,s,l))) = halt SCM+FSA ; :: thesis: LifeSpan (P1,s) <= l
CurInstr (P1,(Comput (P1,s,l))) = CurInstr (P2,(Comput (P2,s,l))) by A1, Lm4, A2, A3;
hence LifeSpan (P1,s) <= l by A5, A7, EXTPRO_1:def 15; :: thesis: verum

end;

CurInstr (P2,(Comput (P2,s,(LifeSpan (P1,s))))) = CurInstr (P1,(Comput (P1,s,(LifeSpan (P1,s))))) by A1, Lm4, A2, A3
.= halt SCM+FSA by A5, EXTPRO_1:def 15 ;
hence LifeSpan (P1,s) = LifeSpan (P2,s) by A6, A4, EXTPRO_1:def 15; :: thesis: Result (P1,s) = Result (P2,s)
thus Result (P1,s) = Result (P2,s) by A1, Th6, A2, A3; :: thesis: verum