let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s being State of SCMPDS st p c= s holds
for i being Element of NAT holds IC (Comput ((ProgramPart s),s,i)) in dom (ProgramPart p)
let s be State of SCMPDS; :: thesis: ( p c= s implies for i being Element of NAT holds IC (Comput ((ProgramPart s),s,i)) in dom (ProgramPart p) )
assume A1: p c= s ; :: thesis: for i being Element of NAT holds IC (Comput ((ProgramPart s),s,i)) in dom (ProgramPart p)
let i be Element of NAT ; :: thesis: IC (Comput ((ProgramPart s),s,i)) in dom (ProgramPart p)
set Csi = Comput ((ProgramPart s),s,i);
set loc = IC (Comput ((ProgramPart s),s,i));
A3: ( IC (Comput ((ProgramPart s),s,i)) in dom (ProgramPart p) iff IC (Comput ((ProgramPart s),s,i)) in (dom p) /\ NAT ) by RELAT_1:90;
assume not IC (Comput ((ProgramPart s),s,i)) in dom (ProgramPart p) ; :: thesis: contradiction
then A4: not IC (Comput ((ProgramPart s),s,i)) in dom p by A3, XBOOLE_0:def 4;
set p2 = p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1));
set p1 = p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0));
A5: dom (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0))) = (dom p) \/ (dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0))) by FUNCT_4:def 1;
A6: dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1)) = {(IC (Comput ((ProgramPart s),s,i)))} by FUNCOP_1:19;
then A7: IC (Comput ((ProgramPart s),s,i)) in dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1)) by TARSKI:def 1;
A8: dom (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1))) = (dom p) \/ (dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1))) by FUNCT_4:def 1;
then A9: IC (Comput ((ProgramPart s),s,i)) in dom (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1))) by A7, XBOOLE_0:def 3;
consider s2 being State of SCMPDS such that
A10: p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1)) c= s2 by PBOOLE:156;
set Cs2i = Comput ((ProgramPart s2),s2,i);
consider s1 being State of SCMPDS such that
A11: p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0)) c= s1 by PBOOLE:156;
set Cs1i = Comput ((ProgramPart s1),s1,i);
A12: dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0)) = {(IC (Comput ((ProgramPart s),s,i)))} by FUNCOP_1:19;
then A13: IC (Comput ((ProgramPart s),s,i)) in dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0)) by TARSKI:def 1;
then A14: IC (Comput ((ProgramPart s),s,i)) in dom (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0))) by A5, XBOOLE_0:def 3;
not p is autonomic

proof

A15: now

let x be set ; :: thesis: ( x in dom p implies p . x = s2 . x )
assume A16: x in dom p ; :: thesis: p . x = s2 . x
dom p misses dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1)) by A4, A6, ZFMISC_1:56;
then A17: p . x = (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1))) . x by A16, FUNCT_4:17;
x in dom (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1))) by A8, A16, XBOOLE_0:def 3;
hence p . x = s2 . x by A10, A17, GRFUNC_1:8; :: thesis: verum

end;

((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1)) . (IC (Comput ((ProgramPart s),s,i))) = goto 1 by FUNCOP_1:87;
then (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 1))) . (IC (Comput ((ProgramPart s),s,i))) = goto 1 by A7, FUNCT_4:14;
then s2 . (IC (Comput ((ProgramPart s),s,i))) = goto 1 by A9, A10, GRFUNC_1:8;
then A18: (Comput ((ProgramPart s2),s2,i)) . (IC (Comput ((ProgramPart s),s,i))) = goto 1 by AMI_1:54;
((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0)) . (IC (Comput ((ProgramPart s),s,i))) = goto 0 by FUNCOP_1:87;
then (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0))) . (IC (Comput ((ProgramPart s),s,i))) = goto 0 by A13, FUNCT_4:14;
then s1 . (IC (Comput ((ProgramPart s),s,i))) = goto 0 by A14, A11, GRFUNC_1:8;
then A19: (Comput ((ProgramPart s1),s1,i)) . (IC (Comput ((ProgramPart s),s,i))) = goto 0 by AMI_1:54;
take s1 ; :: according to EXTPRO_1:def 9 :: thesis: ex b₁ being set st
( p c= s1 & p c= b₁ & not for b₂ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₂)) | (proj1 p) = (Comput ((ProgramPart b₁),b₁,b₂)) | (proj1 p) )
take s2 ; :: thesis: ( p c= s1 & p c= s2 & not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p) )

A20: now

let x be set ; :: thesis: ( x in dom p implies p . x = s1 . x )
assume A21: x in dom p ; :: thesis: p . x = s1 . x
dom p misses dom ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0)) by A4, A12, ZFMISC_1:56;
then A22: p . x = (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0))) . x by A21, FUNCT_4:17;
x in dom (p +* ((IC (Comput ((ProgramPart s),s,i))) .--> (goto 0))) by A5, A21, XBOOLE_0:def 3;
hence p . x = s1 . x by A11, A22, GRFUNC_1:8; :: thesis: verum

end;

dom s1 = the carrier of SCMPDS by PARTFUN1:def 4;
then dom p c= dom s1 by RELAT_1:def 18;
hence A23: p c= s1 by A20, GRFUNC_1:8; :: thesis: ( p c= s2 & not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p) )
then A24: (Comput ((ProgramPart s1),s1,i)) | (dom p) = (Comput ((ProgramPart s),s,i)) | (dom p) by A1, EXTPRO_1:def 9;
dom s2 = the carrier of SCMPDS by PARTFUN1:def 4;
then dom p c= dom s2 by RELAT_1:def 18;
hence p c= s2 by A15, GRFUNC_1:8; :: thesis: not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p)
then A25: (Comput ((ProgramPart s1),s1,i)) | (dom p) = (Comput ((ProgramPart s2),s2,i)) | (dom p) by A23, EXTPRO_1:def 9;
take k = i + 1; :: thesis: not (Comput ((ProgramPart s1),s1,k)) | (proj1 p) = (Comput ((ProgramPart s2),s2,k)) | (proj1 p)
set Cs2k = Comput ((ProgramPart s2),s2,k);
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,i)) by AMI_1:123;
A26: Comput ((ProgramPart s2),s2,k) = 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 ;
Y: (ProgramPart (Comput ((ProgramPart s2),s2,i))) /. (IC (Comput ((ProgramPart s2),s2,i))) = (Comput ((ProgramPart s2),s2,i)) . (IC (Comput ((ProgramPart s2),s2,i))) by COMPOS_1:38;
A27: (Comput ((ProgramPart s),s,i)) . (IC SCMPDS) = ((Comput ((ProgramPart s),s,i)) | (dom p)) . (IC SCMPDS) by Th17, FUNCT_1:72;
then (Comput ((ProgramPart s2),s2,i)) . (IC SCMPDS) = IC (Comput ((ProgramPart s),s,i)) by A24, A25, Th17, FUNCT_1:72;
then A28: (Comput ((ProgramPart s2),s2,k)) . (IC SCMPDS) = ICplusConst ((Comput ((ProgramPart s2),s2,i)),1) by A26, A18, Y, SCMPDS_2:66;
A29: (Comput ((ProgramPart s1),s1,i)) . (IC SCMPDS) = ((Comput ((ProgramPart s1),s1,i)) | (dom p)) . (IC SCMPDS) by Th17, FUNCT_1:72;
then A30: IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) by A25, Th17, FUNCT_1:72;
set Cs1k = Comput ((ProgramPart s1),s1,k);
A31: ( ((Comput ((ProgramPart s1),s1,k)) | (dom p)) . (IC SCMPDS) = (Comput ((ProgramPart s1),s1,k)) . (IC SCMPDS) & ((Comput ((ProgramPart s2),s2,k)) | (dom p)) . (IC SCMPDS) = (Comput ((ProgramPart s2),s2,k)) . (IC SCMPDS) ) by Th17, FUNCT_1:72;
Y: (ProgramPart (Comput ((ProgramPart s1),s1,i))) /. (IC (Comput ((ProgramPart s1),s1,i))) = (Comput ((ProgramPart s1),s1,i)) . (IC (Comput ((ProgramPart s1),s1,i))) by COMPOS_1:38;
T: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,i)) by AMI_1:123;
Comput ((ProgramPart s1),s1,k) = 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 ;
then (Comput ((ProgramPart s1),s1,k)) . (IC SCMPDS) = ICplusConst ((Comput ((ProgramPart s1),s1,i)),0) by A19, A24, A29, A27, Y, SCMPDS_2:66;
hence not (Comput ((ProgramPart s1),s1,k)) | (proj1 p) = (Comput ((ProgramPart s2),s2,k)) | (proj1 p) by A28, A30, A31, Th19; :: thesis: verum

end;

hence contradiction ; :: thesis: verum