let p be autonomic FinPartState of SCM; :: thesis: ( IC SCM in dom p implies IC p in dom p )
assume A1: IC SCM in dom p ; :: thesis: IC p in dom p
set il = IC p;
set p1 = p +* ((IC p) .--> (SCM-goto 0));
set p2 = p +* ((IC p) .--> (SCM-goto 1));
consider s1 being State of SCM such that
A2: p +* ((IC p) .--> (SCM-goto 0)) c= s1 by PBOOLE:156;
consider s2 being State of SCM such that
A3: p +* ((IC p) .--> (SCM-goto 1)) c= s2 by PBOOLE:156;
assume A4: not IC p in dom p ; :: thesis: contradiction
not p is autonomic

proof

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) )
A5: dom ((IC p) .--> (SCM-goto 1)) = {(IC p)} by FUNCOP_1:19;
then A6: IC p in dom ((IC p) .--> (SCM-goto 1)) by TARSKI:def 1;
A7: dom p misses {(IC p)} by A4, ZFMISC_1:56;
then A8: p c= p +* ((IC p) .--> (SCM-goto 1)) by A5, FUNCT_4:33;
dom (p +* ((IC p) .--> (SCM-goto 1))) = (dom p) \/ (dom ((IC p) .--> (SCM-goto 1))) by FUNCT_4:def 1;
then IC p in dom (p +* ((IC p) .--> (SCM-goto 1))) by A6, XBOOLE_0:def 3;
then X: s2 . (IC p) = (p +* ((IC p) .--> (SCM-goto 1))) . (IC p) by A3, GRFUNC_1:8
.= ((IC p) .--> (SCM-goto 1)) . (IC p) by A6, FUNCT_4:14
.= SCM-goto 1 by FUNCOP_1:87 ;
Y: (ProgramPart s2) /. (IC s2) = s2 . (IC s2) by COMPOS_1:38;
p c= s2 by A3, A8, XBOOLE_1:1;
then A9: (Following ((ProgramPart s2),s2)) . (IC SCM) = (Exec ((SCM-goto 1),s2)) . (IC SCM) by A1, X, Y, GRFUNC_1:8
.= 1 by AMI_3:13 ;
A10: dom ((IC p) .--> (SCM-goto 0)) = {(IC p)} by FUNCOP_1:19;
then A11: IC p in dom ((IC p) .--> (SCM-goto 0)) by TARSKI:def 1;
A12: p c= p +* ((IC p) .--> (SCM-goto 0)) by A10, A7, FUNCT_4:33;
hence ( p c= s1 & p c= s2 ) by A2, A3, A8, XBOOLE_1:1; :: thesis: not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p)
take 1 ; :: thesis: not (Comput ((ProgramPart s1),s1,1)) | (proj1 p) = (Comput ((ProgramPart s2),s2,1)) | (proj1 p)
assume A13: (Comput ((ProgramPart s1),s1,1)) | (dom p) = (Comput ((ProgramPart s2),s2,1)) | (dom p) ; :: thesis: contradiction
A14: (Following ((ProgramPart s1),s1)) | (dom p) = (Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,0)))) | (dom p) by EXTPRO_1:3
.= (Comput ((ProgramPart s1),s1,(0 + 1))) | (dom p) by EXTPRO_1:4
.= (Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,0)))) | (dom p) by A13, EXTPRO_1:4
.= (Following ((ProgramPart s2),s2)) | (dom p) by EXTPRO_1:3 ;
dom (p +* ((IC p) .--> (SCM-goto 0))) = (dom p) \/ (dom ((IC p) .--> (SCM-goto 0))) by FUNCT_4:def 1;
then IC p in dom (p +* ((IC p) .--> (SCM-goto 0))) by A11, XBOOLE_0:def 3;
then X: s1 . (IC p) = (p +* ((IC p) .--> (SCM-goto 0))) . (IC p) by A2, GRFUNC_1:8
.= ((IC p) .--> (SCM-goto 0)) . (IC p) by A11, FUNCT_4:14
.= SCM-goto 0 by FUNCOP_1:87 ;
Y: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
p c= s1 by A2, A12, XBOOLE_1:1;
then (Following ((ProgramPart s1),s1)) . (IC SCM) = (Exec ((SCM-goto 0),s1)) . (IC SCM) by A1, X, Y, GRFUNC_1:8
.= 0 by AMI_3:13 ;
then 0 = ((Following ((ProgramPart s1),s1)) | (dom p)) . (IC SCM) by A1, FUNCT_1:72
.= 1 by A1, A9, A14, FUNCT_1:72 ;
hence contradiction ; :: thesis: verum

end;

hence contradiction ; :: thesis: verum