let s be State of SCM+FSA; :: thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let I, J be parahalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let a be read-write Int-Location ; :: thesis: ( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A1: I is_halting_on Initialized s by SCMFSA7B:25;

now

let s be State of SCM+FSA; :: thesis: ( (if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)) c= s implies ProgramPart b₁ halts_on b₁ )
A2: I is_closed_on s by SCMFSA7B:24;
A3: I is_halting_on s by SCMFSA7B:25;
A4: J is_halting_on s by SCMFSA7B:25;
A5: J is_closed_on s by SCMFSA7B:24;
assume (if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)) c= s ; :: thesis: ProgramPart b₁ halts_on b₁
then A6: s = s +* ((if>0 (a,I,J)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:79;

per cases ( s . a > 0 or s . a <= 0 ) ;

suppose s . a > 0 ; :: thesis: ProgramPart b₁ halts_on b₁

then if>0 (a,I,J) is_halting_on s by A2, A3, Th22;
hence ProgramPart s halts_on s by A6, SCMFSA7B:def 8; :: thesis: verum

end;

suppose s . a <= 0 ; :: thesis: ProgramPart b₁ halts_on b₁

then if>0 (a,I,J) is_halting_on s by A5, A4, Th24;
hence ProgramPart s halts_on s by A6, SCMFSA7B:def 8; :: thesis: verum

end;

end;

end;

then (if>0 (a,I,J)) +* (Start-At (0,SCM+FSA)) is halting by EXTPRO_1:def 10;
hence if>0 (a,I,J) is parahalting by SCMFSA6B:def 3; :: thesis: ( ( s . a > 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
I is_closed_on Initialized s by SCMFSA7B:24;
hence ( s . a > 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by A1, Th23; :: thesis: ( s . a <= 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
A7: J is_halting_on Initialized s by SCMFSA7B:25;
J is_closed_on Initialized s by SCMFSA7B:24;
hence ( s . a <= 0 implies IExec ((if>0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by A7, Th25; :: thesis: verum