let s be State of SCM+FSA; :: thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on Initialized s & J is_halting_on Initialized s holds
IExec ((if<0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let I, J be Program of SCM+FSA; :: thesis: for a being read-write Int-Location st s . a > 0 & J is_closed_on Initialized s & J is_halting_on Initialized s holds
IExec ((if<0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let a be read-write Int-Location ; :: thesis: ( s . a > 0 & J is_closed_on Initialized s & J is_halting_on Initialized s implies IExec ((if<0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume A1: s . a > 0 ; :: thesis: ( not J is_closed_on Initialized s or not J is_halting_on Initialized s or IExec ((if<0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume that
A2: J is_closed_on Initialized s and
A3: J is_halting_on Initialized s ; :: thesis: IExec ((if<0 (a,I,J)),s) = (IExec (J,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
A4: (Initialized s) . a > 0 by A1, SCMFSA6C:3;
then A5: if>0 (a,J,I) is_halting_on Initialized s by A2, A3, Th22;
if>0 (a,J,I) is_closed_on Initialized s by A2, A3, A4, Th22;
hence IExec ((if<0 (a,I,J)),s) = (IExec ((if>0 (a,J,I)),s)) +* (Start-At ((((card (if>0 (a,J,I))) + (card J)) + 3),SCM+FSA)) by A1, A5, Th19
.= (IExec ((if>0 (a,J,I)),s)) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by Th15
.= ((IExec (J,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by A1, A2, A3, Th23
.= (IExec (J,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by FUNCT_4:122 ;
:: thesis: verum