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 Th27:
  x in dom (F[:](f,z)) implies (F[:](f,z)).x = F.(f.x,z)
proof
A1: dom <:f, dom f --> z:> = dom f /\ dom (dom f --> z) by FUNCT_3:def 7;
  assume
A2: x in dom (F[:](f,z));
  then x in dom <:f, dom f --> z:> by FUNCT_1:11;
  then
A3: x in dom f by A1;
  thus (F[:](f,z)).x = F.(f.x,(dom f --> z).x) by A2,Lm1
    .= F.(f.x,z) by A3,Th7;
end;
