reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);

theorem
  (multpfunc A).(f,g) = (multpfunc A).(g,f)
proof
A1: dom((multpfunc A).(g,f)) = dom g/\ dom f by Th7;
A2: dom((multpfunc A).(f,g)) = dom f/\ dom g by Th7;
  now
    let x be Element of A;
    assume
A3: x in dom f /\ dom g;
    hence ((multpfunc A).(f,g)).x = g.x * f.x by A2,Th7
      .= ((multpfunc A).(g,f)).x by A1,A3,Th7;
  end;
  hence thesis by A2,A1,PARTFUN1:5;
end;
