
theorem Th30:
for I being really-closed MacroInstruction of SCM+FSA, a being Int-Location,
     k being Nat st k <= card I
  holds a>0_goto k ";" I is really-closed
proof let I be really-closed MacroInstruction of SCM+FSA, a being Int-Location,
     k being Nat such that
A1: k <= card I;
A2: a>0_goto k ";" I = stop(Macro(a>0_goto k) ';' Goto 1) ';' I by Lm4
     .= Macro(a>0_goto k) ';' Macro(goto 1) ';' I by Lm5
     .= Macro(a>0_goto k) ';' (Macro(goto 1) ';' I) by COMPOS_1:29;
A3:  card(Macro(goto 1) ';' I) = card Macro(goto 1) + card I - 1 by COMPOS_1:20
     .= 2 + card I - 1 by COMPOS_1:56;
   card I <= card I + 1 by NAT_1:11;
   then
A4: k <= card(Macro(goto 1) ';' I) by A3,A1,XXREAL_0:2;
   0+1 <= card I by NAT_1:13;
   then Macro(goto 1) ';' I is really-closed by Th24;
  hence a>0_goto k ";" I is really-closed  by A4,A2,Th28;
end;
