reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;

theorem Th32:
  x in dom (F[;](z,f)) implies (F[;](z,f)).x = F.(z,f.x)
proof
A1: dom <:dom f --> z, f:> = dom (dom f --> z) /\ dom f by FUNCT_3:def 7;
  assume
A2: x in dom (F[;](z,f));
  then x in dom <:dom f --> z, f:> by FUNCT_1:11;
  then
A3: x in dom f by A1;
  thus (F[;](z,f)).x = F.((dom f --> z).x, f.x) by A2,Lm1
    .= F.(z, f.x) by A3,Th7;
end;
