reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th4:
  disjoin (x .--> X) = x .--> [:X,{x}:]
proof
A1: dom disjoin ({x} --> X) = dom ({x} --> X) by Def3;
A2: dom ({x} --> X) = {x};
  now
    let y be object;
    assume
A4: y in {x};
    then
A5: disjoin ({x} --> X).y = [:({x} --> X).y,{y}:] by A2,Def3;
A6: ({x} --> X).y = X by A4,FUNCOP_1:7;
    ({x} --> [:X,{x}:]).y = [:X,{x}:] by A4,FUNCOP_1:7;
    hence disjoin ({x} --> X).y = ({x} --> [:X,{x}:]).y by A4,A5,A6,
TARSKI:def 1;
  end;
  hence thesis by A1;
end;
