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;
