reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th53:
  <:pr1(X,Y),pr2(X,Y):> = id [:X,Y:]
proof
  dom pr1(X,Y) = [:X,Y:] & dom pr2(X,Y) = [:X,Y:] by Def4,Def5;
  then
A1: dom <:pr1(X,Y),pr2(X,Y):> = [:X,Y:] by Th50;
A2: for x,y being object st x in X & y in Y
    holds <:pr1(X,Y),pr2(X,Y):>.(x,y) = (id [:X,Y:]).(x,y)
  proof
    let x,y be object;
    assume
A3: x in X & y in Y;
    then
A4: [x,y] in [:X,Y:] by ZFMISC_1:87;
    hence <:pr1(X,Y),pr2(X,Y):>.(x,y) = [pr1(X,Y).(x,y),pr2(X,Y).(x,y)] by A1
,Def7
      .= [x,pr2(X,Y).(x,y)] by A3,Def4
      .= [x,y] by A3,Def5
      .= (id [:X,Y:]).(x,y) by A4,FUNCT_1:18;
  end;
  dom id [:X,Y:] = [:X,Y:];
  hence thesis by A1,A2,Th6;
end;
