let s1, s2 be State of SCMPDS; :: thesis: for I being Program of SCMPDS st I is_closed_on s1 & I is_halting_on s1 & Initialize (stop I) c= s1 & Initialize (stop I) c= s2 & ex k being Element of NAT st Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT holds
Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT
let I be Program of SCMPDS; :: thesis: ( I is_closed_on s1 & I is_halting_on s1 & Initialize (stop I) c= s1 & Initialize (stop I) c= s2 & ex k being Element of NAT st Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT implies Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT )
set pI = stop I;
assume A1: I is_closed_on s1 ; :: thesis: ( not I is_halting_on s1 or not Initialize (stop I) c= s1 or not Initialize (stop I) c= s2 or for k being Element of NAT holds not Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT or Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT )
I1: s1 +* (Initialize (stop I)) = (Initialize s1) +* (stop I) by COMPOS_1:125;
assume A2: I is_halting_on s1 ; :: thesis: ( not Initialize (stop I) c= s1 or not Initialize (stop I) c= s2 or for k being Element of NAT holds not Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT or Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT )
assume A3: Initialize (stop I) c= s1 ; :: thesis: ( not Initialize (stop I) c= s2 or for k being Element of NAT holds not Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT or Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT )
then A4: s1 = (Initialize s1) +* (stop I) by I1, FUNCT_4:79;
then A5: ProgramPart s1 halts_on s1 by A2, SCMPDS_6:def 3;
then consider n being Element of NAT such that
A6: CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,n))) = halt SCMPDS by EXTPRO_1:30;
I2: s2 +* (Initialize (stop I)) = (Initialize s2) +* (stop I) by COMPOS_1:125;
assume Initialize (stop I) c= s2 ; :: thesis: ( for k being Element of NAT holds not Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT or Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT )
then A7: s2 = (Initialize s2) +* (stop I) by I2, FUNCT_4:79;
given k being Element of NAT such that A8: Comput ((ProgramPart s1),s1,k),s2 equal_outside NAT ; :: thesis: Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT
set s3 = Comput ((ProgramPart s1),s1,k);
A9: IC SCMPDS in dom (Comput ((ProgramPart s1),s1,k)) by COMPOS_1:9;
I3: (Comput ((ProgramPart s1),s1,k)) +* (Initialize (stop I)) = (Initialize (Comput ((ProgramPart s1),s1,k))) +* (stop I) by COMPOS_1:125;
stop I c= Initialize (stop I) by COMPOS_1:126;
then stop I c= s1 by A3, XBOOLE_1:1;
then A10: stop I c= Comput ((ProgramPart s1),s1,k) by AMI_1:86;
IC (Comput ((ProgramPart s1),s1,k)) = IC ((Initialize s2) +* (stop I)) by A8, A7, COMPOS_1:24
.= 0 by SCMPDS_6:21 ;
then (IC SCMPDS) .--> 0 c= Comput ((ProgramPart s1),s1,k) by A9, FUNCOP_1:88;
then (stop I) +* (Start-At (0,SCMPDS)) c= Comput ((ProgramPart s1),s1,k) by A10, FUNCT_4:92;
then A11: Comput ((ProgramPart s1),s1,k) = (Initialize (Comput ((ProgramPart s1),s1,k))) +* (stop I) by I3, FUNCT_4:79;

A12: now

let n be Element of NAT ; :: thesis: IC (Comput ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k)),n)) in dom (stop I)
S: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,k)) by AMI_1:123;
IC (Comput ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k)),n)) = IC (Comput ((ProgramPart s1),s1,(k + n))) by EXTPRO_1:5;
hence IC (Comput ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k)),n)) in dom (stop I) by A1, A4, S, SCMPDS_6:def 2; :: thesis: verum

end;

T: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,k)) by AMI_1:123;
x: Comput ((ProgramPart s1),s1,(k + n)) = Comput ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k)),n) by EXTPRO_1:5;
U: Comput ((ProgramPart s1),s1,(k + n)) = Comput ((ProgramPart s1),s1,n) by A6, EXTPRO_1:6, NAT_1:11;
CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k)),n))) = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,(k + n)))) by x, T
.= CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,n))) by U ;
then ProgramPart (Comput ((ProgramPart s1),s1,k)) halts_on Comput ((ProgramPart s1),s1,k) by A6, EXTPRO_1:30;
then A13: I is_halting_on Comput ((ProgramPart s1),s1,k) by A11, SCMPDS_6:def 3;
A14: DataPart (Comput ((ProgramPart s1),s1,k)) = DataPart s2 by A8, SCMPDS_6:4;
consider k being Element of NAT such that
A15: CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) = halt SCMPDS by A5, EXTPRO_1:30;
Y: (ProgramPart (Comput ((ProgramPart s1),s1,k))) /. (IC (Comput ((ProgramPart s1),s1,k))) = (Comput ((ProgramPart s1),s1,k)) . (IC (Comput ((ProgramPart s1),s1,k))) by COMPOS_1:38;
TX: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,k)) by AMI_1:123;
A16: (ProgramPart s1) . (IC (Comput ((ProgramPart s1),s1,k))) = s1 . (IC (Comput ((ProgramPart s1),s1,k))) by COMPOS_1:2
.= halt SCMPDS by A15, Y, TX, AMI_1:54 ;
TX: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,k)) by AMI_1:123;
I is_closed_on Comput ((ProgramPart s1),s1,k) by A11, A12, SCMPDS_6:def 2;
then Result ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k))), Result ((ProgramPart s2),s2) equal_outside NAT by A7, A14, A11, A13, Th29;
hence Result ((ProgramPart s1),s1), Result ((ProgramPart s2),s2) equal_outside NAT by A16, TX, EXTPRO_1:9; :: thesis: verum