let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated steady-programmed definite AMI-Struct of N
for p being NAT -defined FinPartState of
for s being State of S st p c= s holds
for k being Element of NAT holds p c= Comput (ProgramPart s),s,k
let S be non empty stored-program IC-Ins-separated steady-programmed definite AMI-Struct of N; :: thesis: for p being NAT -defined FinPartState of
for s being State of S st p c= s holds
for k being Element of NAT holds p c= Comput (ProgramPart s),s,k
let p be NAT -defined FinPartState of ; :: thesis: for s being State of S st p c= s holds
for k being Element of NAT holds p c= Comput (ProgramPart s),s,k
let s be State of S; :: thesis: ( p c= s implies for k being Element of NAT holds p c= Comput (ProgramPart s),s,k )
assume A1: p c= s ; :: thesis: for k being Element of NAT holds p c= Comput (ProgramPart s),s,k
let k be Element of NAT ; :: thesis: p c= Comput (ProgramPart s),s,k
A2: dom p c= NAT by RELAT_1:def 18;

A3: now

let x be set ; :: thesis: ( x in dom p implies p . x = (Comput (ProgramPart s),s,k) . x )
assume A4: x in dom p ; :: thesis: p . x = (Comput (ProgramPart s),s,k) . x
then reconsider l = x as Element of NAT by A2;
s . x = (Comput (ProgramPart s),s,k) . l by Th54;
hence p . x = (Comput (ProgramPart s),s,k) . x by A1, A4, GRFUNC_1:8; :: thesis: verum

end;

dom s = the carrier of S by PARTFUN1:def 4
.= dom (Comput (ProgramPart s),s,k) by PARTFUN1:def 4 ;
then dom p c= dom (Comput (ProgramPart s),s,k) by A1, GRFUNC_1:8;
hence p c= Comput (ProgramPart s),s,k by A3, GRFUNC_1:8; :: thesis: verum