let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCMPDS st p c= s1 & p c= s2 holds
for i, m being Element of NAT
for a being Int_position
for k1, k2 being Integer st CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 & m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 holds
( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 )
let s1, s2 be State of SCMPDS; :: thesis: ( p c= s1 & p c= s2 implies for i, m being Element of NAT
for a being Int_position
for k1, k2 being Integer st CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 & m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 holds
( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 ) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i, m being Element of NAT
for a being Int_position
for k1, k2 being Integer st CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 & m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 holds
( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 )
let i, m be Element of NAT ; :: thesis: for a being Int_position
for k1, k2 being Integer st CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 & m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 holds
( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 )
let a be Int_position ; :: thesis: for k1, k2 being Integer st CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 & m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 holds
( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 )
let k1, k2 be Integer; :: thesis: ( CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 & m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 implies ( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 ) )
set Cs1i = Comput ((ProgramPart s1),s1,i);
set Cs2i = Comput ((ProgramPart s2),s2,i);
set Cs1i1 = Comput ((ProgramPart s1),s1,(i + 1));
set Cs2i1 = Comput ((ProgramPart s2),s2,(i + 1));
A2: ( IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) & (Comput ((ProgramPart s1),s1,(i + 1))) | (dom p) = (Comput ((ProgramPart s2),s2,(i + 1))) | (dom p) ) by A1, Th23, EXTPRO_1:def 9;
set I = CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i)));
T: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,i)) by AMI_1:123;
A3: Comput ((ProgramPart s1),s1,(i + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i)))),(Comput ((ProgramPart s1),s1,i))) by T ;
A4: m + 1 >= 0 by NAT_1:2;
A5: ( ((Comput ((ProgramPart s1),s1,(i + 1))) | (dom p)) . (IC SCMPDS) = (Comput ((ProgramPart s1),s1,(i + 1))) . (IC SCMPDS) & ((Comput ((ProgramPart s2),s2,(i + 1))) | (dom p)) . (IC SCMPDS) = (Comput ((ProgramPart s2),s2,(i + 1))) . (IC SCMPDS) ) by Th17, FUNCT_1:72;
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,i)) by AMI_1:123;
A6: Comput ((ProgramPart s2),s2,(i + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,i))),(Comput ((ProgramPart s2),s2,i)))),(Comput ((ProgramPart s2),s2,i))) by T ;
assume that
A7: CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = (a,k1) >=0_goto k2 and
A8: ( m = IC (Comput ((ProgramPart s1),s1,i)) & m + k2 >= 0 & k2 <> 1 ) ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 )
A9: CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i))) = CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,i))),(Comput ((ProgramPart s2),s2,i))) by A1, Th23;

A10: now

assume that
A11: (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 and
A12: (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) >= 0 ; :: thesis: contradiction
A13: (Comput ((ProgramPart s1),s1,(i + 1))) . (IC SCMPDS) = ICplusConst ((Comput ((ProgramPart s1),s1,i)),k2) by A3, A7, A12, SCMPDS_2:69;
(Comput ((ProgramPart s2),s2,(i + 1))) . (IC SCMPDS) = succ (IC (Comput ((ProgramPart s2),s2,i))) by A9, A6, A7, A11, SCMPDS_2:69
.= ICplusConst ((Comput ((ProgramPart s2),s2,i)),1) by Th20 ;
hence contradiction by A5, A2, A8, A4, A13, Th18; :: thesis: verum

end;

now

assume that
A14: (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 and
A15: (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) >= 0 ; :: thesis: contradiction
A16: (Comput ((ProgramPart s2),s2,(i + 1))) . (IC SCMPDS) = ICplusConst ((Comput ((ProgramPart s2),s2,i)),k2) by A9, A6, A7, A15, SCMPDS_2:69;
(Comput ((ProgramPart s1),s1,(i + 1))) . (IC SCMPDS) = succ (IC (Comput ((ProgramPart s1),s1,i))) by A3, A7, A14, SCMPDS_2:69
.= ICplusConst ((Comput ((ProgramPart s1),s1,i)),1) by Th20 ;
hence contradiction by A5, A2, A8, A4, A16, Th18; :: thesis: verum

end;

hence ( (Comput ((ProgramPart s1),s1,i)) . (DataLoc (((Comput ((ProgramPart s1),s1,i)) . a),k1)) < 0 iff (Comput ((ProgramPart s2),s2,i)) . (DataLoc (((Comput ((ProgramPart s2),s2,i)) . a),k1)) < 0 ) by A10; :: thesis: verum