reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;

theorem Th2:
  <:f,g:> = <:g,f:>~
proof
A1: dom <:f,g:> = dom g /\ dom f by FUNCT_3:def 7
    .= dom <:g,f:> by FUNCT_3:def 7;
A2: now
    let x be object;
    assume
A3: x in dom <:f,g:>;
    then
A4: <:g,f:>.x = [g.x, f.x] by A1,FUNCT_3:def 7;
    thus <:f,g:>.x = [f.x, g.x] by A3,FUNCT_3:def 7
      .= <:g,f:>~.x by A1,A3,A4,Def1;
  end;
  dom <:f,g:> = dom (<:g,f:>~) by A1,Def1;
  hence thesis by A2;
end;
