reserve x,N for set,
        k for Nat;
reserve N for with_zero set;
reserve S for IC-Ins-separated non empty with_non-empty_values
     Mem-Struct over N;
reserve s for State of S;
reserve p for PartState of S;

theorem Th29:
  for S be IC-Ins-separated non empty with_non-empty_values Mem-Struct over N,
      l be Nat
  for p be PartState of S
  holds p is l-started iff Start-At(l,S) c= p
proof
 let S be IC-Ins-separated non empty with_non-empty_values Mem-Struct over N,
     l be Nat;
 let p be PartState of S;
 thus p is l-started implies Start-At(l,S) c= p
  proof assume
A2:  p is l-started;
    IC S in dom p by A2;
    then
A3:  dom Start-At(l,S) c= dom p by ZFMISC_1:31;
    for x being object st x in dom Start-At(l,S) holds Start-At(l,S).x = p.x
     proof let x be object;
      assume
A4:     x in dom Start-At(l,S);
      hence Start-At(l,S).x = IC Start-At(l,S) by TARSKI:def 1
         .= l by FUNCOP_1:72
         .= IC p by A2
         .= p.x  by A4,TARSKI:def 1;
     end;
   hence Start-At(l,S) c= p by A3,GRFUNC_1:2;
  end;
 assume
A5: Start-At(l,S) c= p;
  then
A6:  dom Start-At(l,S) c= dom p by RELAT_1:11;
A7: IC S in dom Start-At(l,S) by TARSKI:def 1;
 hence IC S in dom p by A6;
 thus IC p = IC Start-At(l,S) by A5,A7,GRFUNC_1:2
     .= l by FUNCOP_1:72;
end;
