let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being parahalting Program of SCMPDS st stop I c= P1 & stop I c= P2 holds
( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of SCMPDS st stop I c= P1 & stop I c= P2 holds
( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
let I be parahalting Program of SCMPDS; :: thesis: ( stop I c= P1 & stop I c= P2 implies ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) ) )
set SI = stop I;
assume that
A1: stop I c= P1 and
A2: stop I c= P2 ; :: thesis: ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
A4: P2 halts_on s by A2, SCMPDS_4:def 7;
A5: P1 halts_on s by A1, SCMPDS_4:def 7;

A6: now

let l be Element of NAT ; :: thesis: ( CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS implies LifeSpan (P1,s) <= l )
assume A7: CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS ; :: thesis: LifeSpan (P1,s) <= l
CurInstr (P1,(Comput (P1,s,l))) = CurInstr (P2,(Comput (P2,s,l))) by A1, A2, Th20;
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, A2, Th20
.= halt SCMPDS by A5, EXTPRO_1:def 15 ;
hence K: LifeSpan (P1,s) = LifeSpan (P2,s) by A6, A4, EXTPRO_1:def 15; :: thesis: Result (P1,s) = Result (P2,s)
P2 halts_on s by A2, SCMPDS_4:def 7;
then A8: Result (P2,s) = Comput (P2,s,(LifeSpan (P1,s))) by K, EXTPRO_1:23;
P1 halts_on s by A1, SCMPDS_4:def 7;
then Result (P1,s) = Comput (P1,s,(LifeSpan (P1,s))) by EXTPRO_1:23;
hence Result (P1,s) = Result (P2,s) by A1, A2, A8, Th20; :: thesis: verum