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);
reserve L for non empty LattStr,
        p,q,r for Element of L;
reserve p,q,r for Element of RealFunc_Lattice(A);

theorem
  (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) =(maxfuncreal(A)).(((
  maxfuncreal(A)).(q,r)),p) & (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) =(
  maxfuncreal(A)).(((maxfuncreal(A)).(p,q)),r) & (maxfuncreal(A)).(p,((
  maxfuncreal(A)).(q,r))) =(maxfuncreal(A)).(((maxfuncreal(A)).(q,p)),r) & (
maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) =(maxfuncreal(A)).(((maxfuncreal(A
)).(r,p)),q) & (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) =(maxfuncreal(A)).
(((maxfuncreal(A)).(r,q)),p) & (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) =(
  maxfuncreal(A)).(((maxfuncreal(A)).(p,r)),q)
proof
  set s=q"\/"r;
  thus
A1: (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) = s"\/"p by LATTICES:def 1
    .= (maxfuncreal(A)).(((maxfuncreal(A)).(q,r)),p);
  thus (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) = (maxfuncreal(A)).((
  maxfuncreal(A)).(p,q),r) by Th10;
  thus (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) = p"\/"s
    .= (q"\/"p)"\/"r by Lm7
    .= (maxfuncreal(A)).((maxfuncreal(A)).(q,p),r);
  thus
A2: (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r))) = p"\/"(q"\/"r)
    .= (r"\/"p)"\/"q by Lm7
    .= (maxfuncreal(A)).((maxfuncreal(A)).(r,p),q);
  thus (maxfuncreal(A)).(p,((maxfuncreal(A)).(q,r)))= (maxfuncreal(A)).(((
  maxfuncreal(A)).(r,q)),p) by A1,Th21;
  thus thesis by A2,Th21;
end;
