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 Th68:
  for f,g being Function st dom f = X & dom g = X holds <:f,g:> =
  [:f,g:]*(delta X)
proof
  let f,g be Function such that
A1: dom f = X & dom g = X;
A2: dom delta X = X by Def6;
  rng delta X c= [:X,X:] by Th47;
  then rng delta X c= dom [:f,g:] by A1,Def8;
  then
A3: dom([:f,g:]*(delta X)) = X by A2,RELAT_1:27;
  dom f /\ dom g = X by A1;
  then
A4: dom <:f,g:> = X by Def7;
  for x being object st x in X holds <:f,g:>.x = ([:f,g:]*(delta X)).x
  proof
    let x be object;
    assume
A5: x in X;
    hence <:f,g:>.x = [f.x,g.x] by A4,Def7
      .= [:f,g:].(x,x) by A1,A5,Def8
      .= [:f,g:].((delta X).x) by A5,Def6
      .= ([:f,g:]*(delta X)).x by A3,A5,FUNCT_1:12;
  end;
  hence thesis by A4,A3;
end;
