reserve x,y for Real;
reserve a,b,c for Element of Real_Lattice;
reserve p,q,r for Element of Real_Lattice;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);

theorem Th8:
  (maxfuncreal(A)).(f,g) = (maxfuncreal(A)).(g,f)
proof
  now
    let x be Element of A;
A1: x in dom (maxreal.:(f,g)) by Lm6;
A2: x in dom (maxreal.:(g,f)) by Lm6;
    thus ((maxfuncreal(A)).(f,g)).x = (maxreal.:(f,g)).x by Def4
      .= maxreal.(f.x,g.x) by A1,FUNCOP_1:22
      .= maxreal.(g.x,f.x) by Th1
      .= (maxreal.:(g,f)).x by A2,FUNCOP_1:22
      .= ((maxfuncreal(A)).(g,f)).x by Def4;
  end;
  hence thesis by FUNCT_2:63;
end;
