let p be autonomic FinPartState of ; :: thesis: ( IC SCMPDS in dom p implies IC p in dom p )
assume A1: IC SCMPDS in dom p ; :: thesis: IC p in dom p
set il = IC p;
set p1 = p +* ((IC p) .--> (goto 0 ));
set p2 = p +* ((IC p) .--> (goto 1));
consider s1 being State of such that
A2: p +* ((IC p) .--> (goto 0 )) c= s1 by CARD_3:97;
consider s2 being State of such that
A3: p +* ((IC p) .--> (goto 1)) c= s2 by CARD_3:97;
assume A4: not IC p in dom p ; :: thesis: contradiction
not p is autonomic

proof

take s1 ; :: according to AMI_1:def 25 :: thesis: ex b₁ being Element of product the Object-Kind of SCMPDS st
( p c= s1 & p c= b₁ & not for b₂ being Element of NAT holds (Computation s1,b₂) | (dom p) = (Computation b₁,b₂) | (dom p) )
take s2 ; :: thesis: ( p c= s1 & p c= s2 & not for b₁ being Element of NAT holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom p) )
A5: dom ((IC p) .--> (goto 1)) = {(IC p)} by FUNCOP_1:19;
then A6: IC p in dom ((IC p) .--> (goto 1)) by TARSKI:def 1;
A7: dom ((IC p) .--> (goto 0 )) = {(IC p)} by FUNCOP_1:19;
then A8: IC p in dom ((IC p) .--> (goto 0 )) by TARSKI:def 1;
A9: dom p misses {(IC p)} by A4, ZFMISC_1:56;
then A10: p c= p +* ((IC p) .--> (goto 0 )) by A7, FUNCT_4:33;
A11: p c= p +* ((IC p) .--> (goto 1)) by A5, A9, FUNCT_4:33;
hence ( p c= s1 & p c= s2 ) by A2, A3, A10, XBOOLE_1:1; :: thesis: not for b₁ being Element of NAT holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom p)
dom (p +* ((IC p) .--> (goto 1))) = (dom p) \/ (dom ((IC p) .--> (goto 1))) by FUNCT_4:def 1;
then IC p in dom (p +* ((IC p) .--> (goto 1))) by A6, XBOOLE_0:def 3;
then s2 . (IC p) = (p +* ((IC p) .--> (goto 1))) . (IC p) by A3, GRFUNC_1:8
.= ((IC p) .--> (goto 1)) . (IC p) by A6, FUNCT_4:14
.= goto 1 by FUNCOP_1:87 ;
then A12: (Following s2) . (IC SCMPDS ) = (Exec (goto 1),s2) . (IC SCMPDS ) by A1, A3, A11, AMI_1:97, XBOOLE_1:1
.= ICplusConst s2,1 by SCMPDS_2:66 ;
take 1 ; :: thesis: not (Computation s1,1) | (dom p) = (Computation s2,1) | (dom p)
assume A13: (Computation s1,1) | (dom p) = (Computation s2,1) | (dom p) ; :: thesis: contradiction
A14: (Following s1) | (dom p) = (Following (Computation s1,0 )) | (dom p) by AMI_1:13
.= (Computation s1,(0 + 1)) | (dom p) by AMI_1:14
.= (Following (Computation s2,0 )) | (dom p) by A13, AMI_1:14
.= (Following s2) | (dom p) by AMI_1:13 ;
dom (p +* ((IC p) .--> (goto 0 ))) = (dom p) \/ (dom ((IC p) .--> (goto 0 ))) by FUNCT_4:def 1;
then IC p in dom (p +* ((IC p) .--> (goto 0 ))) by A8, XBOOLE_0:def 3;
then s1 . (IC p) = (p +* ((IC p) .--> (goto 0 ))) . (IC p) by A2, GRFUNC_1:8
.= ((IC p) .--> (goto 0 )) . (IC p) by A8, FUNCT_4:14
.= goto 0 by FUNCOP_1:87 ;
then (Following s1) . (IC SCMPDS ) = (Exec (goto 0 ),s1) . (IC SCMPDS ) by A1, A2, A10, AMI_1:97, XBOOLE_1:1
.= ICplusConst s1,0 by SCMPDS_2:66 ;
then A15: ICplusConst s1,0 = ((Following s1) | (dom p)) . (IC SCMPDS ) by A1, FUNCT_1:72
.= ICplusConst s2,1 by A1, A12, A14, FUNCT_1:72 ;
IC s2 = IC p by A1, A3, A11, AMI_1:97, XBOOLE_1:1
.= IC s1 by A1, A2, A10, AMI_1:97, XBOOLE_1:1 ;
hence contradiction by A15, Th19; :: thesis: verum

end;

hence contradiction ; :: thesis: verum