let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s1, s2 being State of SCM+FSA
for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= P1 & I c= P2 & s1 = s2 holds
( LifeSpan (P1,s1) = LifeSpan (P2,s2) & Result (P1,s1) = Result (P2,s2) )
let s1, s2 be State of SCM+FSA; :: thesis: for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= P1 & I c= P2 & s1 = s2 holds
( LifeSpan (P1,s1) = LifeSpan (P2,s2) & Result (P1,s1) = Result (P2,s2) )
let I be InitHalting Program of SCM+FSA; :: thesis: ( Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= P1 & I c= P2 & s1 = s2 implies ( LifeSpan (P1,s1) = LifeSpan (P2,s2) & Result (P1,s1) = Result (P2,s2) ) )
assume that
A1: Initialize ((intloc 0) .--> 1) c= s1 and
A2: Initialize ((intloc 0) .--> 1) c= s2 and
A3: I c= P1 and
A4: I c= P2 and
A5: s1 = s2 ; :: thesis: ( LifeSpan (P1,s1) = LifeSpan (P2,s2) & Result (P1,s1) = Result (P2,s2) )
A6: P2 halts_on s2 by A2, A4, SCM_HALT:def 2;
A7: P1 halts_on s1 by A1, A3, SCM_HALT:def 2;

A8: now

let l be Element of NAT ; :: thesis: ( CurInstr (P2,(Comput (P2,s2,l))) = halt SCM+FSA implies LifeSpan (P1,s1) <= l )
assume A9: CurInstr (P2,(Comput (P2,s2,l))) = halt SCM+FSA ; :: thesis: LifeSpan (P1,s1) <= l
CurInstr (P1,(Comput (P1,s1,l))) = CurInstr (P2,(Comput (P2,s2,l))) by A1, A5, Th11, A3, A4;
hence LifeSpan (P1,s1) <= l by A7, A9, EXTPRO_1:def 15; :: thesis: verum

end;

CurInstr (P2,(Comput (P2,s2,(LifeSpan (P1,s1))))) = CurInstr (P1,(Comput (P1,s1,(LifeSpan (P1,s1))))) by A1, A5, Th11, A3, A4
.= halt SCM+FSA by A7, EXTPRO_1:def 15 ;
hence A10: LifeSpan (P1,s1) = LifeSpan (P2,s2) by A8, A6, EXTPRO_1:def 15; :: thesis: Result (P1,s1) = Result (P2,s2)
P2 halts_on s2 by A2, A4, SCM_HALT:def 2;
then A11: Result (P2,s2) = Comput (P2,s2,(LifeSpan (P1,s1))) by A10, EXTPRO_1:23;
P1 halts_on s1 by A1, A3, SCM_HALT:def 2;
then Result (P1,s1) = Comput (P1,s1,(LifeSpan (P1,s1))) by EXTPRO_1:23;
hence Result (P1,s1) = Result (P2,s2) by A1, A5, A11, Th11, A3, A4; :: thesis: verum