reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem Th54:
  [[x,x9],[y,y9]] in dom |:f,g:| iff [x,y] in dom f & [x9,y9] in dom g
proof
  thus [[x,x9],[y,y9]] in dom |:f,g:| implies [x,y] in dom f & [x9,y9] in dom
  g
  proof
    assume [[x,x9],[y,y9]] in dom |:f,g:|;
    then consider x1,y1,x19,y19 being object such that
A1: [[x,x9],[y,y9]] = [[x1,x19],[y1,y19]] and
A2: [x1,y1] in dom f & [x19,y19] in dom g by Def3;
    x = x1 & x9= x19 by A1,Lm1;
    hence thesis by A1,A2,Lm1;
  end;
  thus thesis by Def3;
end;
