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 Th9:
  (minfuncreal(A)).(f,g) = (minfuncreal(A)).(g,f)
proof
  now
    let x be Element of A;
A1: x in dom (minreal.:(f,g)) by Lm6;
A2: x in dom (minreal.:(g,f)) by Lm6;
    thus ((minfuncreal(A)).(f,g)).x = (minreal.:(f,g)).x by Def5
      .= minreal.(f.x,g.x) by A1,FUNCOP_1:22
      .= minreal.(g.x,f.x) by Th2
      .= (minreal.:(g,f)).x by A2,FUNCOP_1:22
      .= ((minfuncreal(A)).(g,f)).x by Def5;
  end;
  hence thesis by FUNCT_2:63;
end;
