let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s1, s2 being 0 -started State of SCMPDS
for I being Program of SCMPDS st I is_closed_on s1,P1 & I is_halting_on s1,P1 & stop I c= P1 & stop I c= P2 & ex k being Element of NAT st Comput (P1,s1,k) = s2 holds
Result (P1,s1) = Result (P2,s2)
let s1, s2 be 0 -started State of SCMPDS; :: thesis: for I being Program of SCMPDS st I is_closed_on s1,P1 & I is_halting_on s1,P1 & stop I c= P1 & stop I c= P2 & ex k being Element of NAT st Comput (P1,s1,k) = s2 holds
Result (P1,s1) = Result (P2,s2)
let I be Program of SCMPDS; :: thesis: ( I is_closed_on s1,P1 & I is_halting_on s1,P1 & stop I c= P1 & stop I c= P2 & ex k being Element of NAT st Comput (P1,s1,k) = s2 implies Result (P1,s1) = Result (P2,s2) )
set pI = stop I;
assume A1: I is_closed_on s1,P1 ; :: thesis: ( not I is_halting_on s1,P1 or not stop I c= P1 or not stop I c= P2 or for k being Element of NAT holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
assume A3: I is_halting_on s1,P1 ; :: thesis: ( not stop I c= P1 or not stop I c= P2 or for k being Element of NAT holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
assume stop I c= P1 ; :: thesis: ( not stop I c= P2 or for k being Element of NAT holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
then A5: P1 = P1 +* (stop I) by FUNCT_4:98;
assume stop I c= P2 ; :: thesis: ( for k being Element of NAT holds not Comput (P1,s1,k) = s2 or Result (P1,s1) = Result (P2,s2) )
then XX: P2 = P2 +* (stop I) by FUNCT_4:98;
A2: s1 = Initialize s1 by MEMSTR_0:44;
then A6: P1 halts_on s1 by A3, A5, SCMPDS_6:def 3;
then consider n being Element of NAT such that
A7: CurInstr (P1,(Comput (P1,s1,n))) = halt SCMPDS by EXTPRO_1:29;
A9: s2 = Initialize s2 by MEMSTR_0:44;
given k being Element of NAT such that A10: Comput (P1,s1,k) = s2 ; :: thesis: Result (P1,s1) = Result (P2,s2)
set s3 = Comput (P1,s1,k);
set P3 = P1;
A11: IC in dom (Comput (P1,s1,k)) by MEMSTR_0:2;
IC (Comput (P1,s1,k)) = 0 by MEMSTR_0:def 8, A10;
then Start-At (0,SCMPDS) c= Comput (P1,s1,k) by A11, FUNCOP_1:73;
then A14: Comput (P1,s1,k) = Initialize (Comput (P1,s1,k)) by FUNCT_4:98;

A15: now

let n be Element of NAT ; :: thesis: IC (Comput (P1,(Comput (P1,s1,k)),n)) in dom (stop I)
IC (Comput (P1,(Comput (P1,s1,k)),n)) = IC (Comput (P1,s1,(k + n))) by EXTPRO_1:4;
hence IC (Comput (P1,(Comput (P1,s1,k)),n)) in dom (stop I) by A1, A5, A2, SCMPDS_6:def 2; :: thesis: verum

end;

CurInstr (P1,(Comput (P1,(Comput (P1,s1,k)),n))) = CurInstr (P1,(Comput (P1,s1,(k + n)))) by EXTPRO_1:4
.= CurInstr (P1,(Comput (P1,s1,n))) by A7, EXTPRO_1:5, NAT_1:11 ;
then P1 halts_on Comput (P1,s1,k) by A7, EXTPRO_1:29;
then A20: I is_halting_on Comput (P1,s1,k),P1 by A14, A5, SCMPDS_6:def 3;
A21: DataPart (Comput (P1,s1,k)) = DataPart s2 by A10;
consider k being Element of NAT such that
A22: CurInstr (P1,(Comput (P1,s1,k))) = halt SCMPDS by A6, EXTPRO_1:29;
A25: P1 . (IC (Comput (P1,s1,k))) = halt SCMPDS by A22, PBOOLE:143;
I is_closed_on Comput (P1,s1,k),P1 by A14, A15, A5, SCMPDS_6:def 2;
then Result (P1,(Comput (P1,s1,k))) = Result (P2,s2) by A9, A21, A14, A20, Th29, XX, A5;
hence Result (P1,s1) = Result (P2,s2) by A25, EXTPRO_1:8; :: thesis: verum