let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s1, s2 being State of SCMPDS
for I being Program of st I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 holds
( LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) & Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) )
let s1, s2 be State of SCMPDS; :: thesis: for I being Program of st I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 holds
( LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) & Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) )
let I be Program of ; :: thesis: ( I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 implies ( LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) & Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) ) )
assume A1: I is_closed_on s1,P1 ; :: thesis: ( not I is_halting_on s1,P1 or not DataPart s1 = DataPart s2 or ( LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) & Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) ) )
set ss1 = Initialize s1;
set PP1 = P1 +* (stop I);
set ss2 = Initialize s2;
set PP2 = P2 +* (stop I);
assume A2: I is_halting_on s1,P1 ; :: thesis: ( not DataPart s1 = DataPart s2 or ( LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) & Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) ) )
then A3: P1 +* (stop I) halts_on Initialize s1 by SCMPDS_6:def 3;
then A4: Result ((P1 +* (stop I)),(Initialize s1)) = Comput ((P1 +* (stop I)),(Initialize s1),(LifeSpan ((P1 +* (stop I)),(Initialize s1)))) by EXTPRO_1:23;
assume A5: DataPart s1 = DataPart s2 ; :: thesis: ( LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) & Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) )
then I is_halting_on s2,P2 by A1, A2, SCMPDS_6:23;
then A6: P2 +* (stop I) halts_on Initialize s2 by SCMPDS_6:def 3;

A7: now :: thesis: for l being Nat st CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),l))) = halt SCMPDS holds
LifeSpan ((P1 +* (stop I)),(Initialize s1)) <= l

let l be Nat; :: thesis: ( CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),l))) = halt SCMPDS implies LifeSpan ((P1 +* (stop I)),(Initialize s1)) <= l )
assume A8: CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),l))) = halt SCMPDS ; :: thesis: LifeSpan ((P1 +* (stop I)),(Initialize s1)) <= l
CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),l))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),l))) by A1, A5, Th6;
hence LifeSpan ((P1 +* (stop I)),(Initialize s1)) <= l by A3, A8, EXTPRO_1:def 15; :: thesis: verum

end;

CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),(LifeSpan ((P1 +* (stop I)),(Initialize s1)))))) = CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),(LifeSpan ((P1 +* (stop I)),(Initialize s1)))))) by A1, A5, Th6
.= halt SCMPDS by A3, EXTPRO_1:def 15 ;
hence LifeSpan ((P1 +* (stop I)),(Initialize s1)) = LifeSpan ((P2 +* (stop I)),(Initialize s2)) by A7, A6, EXTPRO_1:def 15; :: thesis: Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2))
then Result ((P2 +* (stop I)),(Initialize s2)) = Comput ((P2 +* (stop I)),(Initialize s2),(LifeSpan ((P1 +* (stop I)),(Initialize s1)))) by A6, EXTPRO_1:23;
hence Result ((P1 +* (stop I)),(Initialize s1)) = Result ((P2 +* (stop I)),(Initialize s2)) by A1, A5, A4, Th6; :: thesis: verum