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
  (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) =(minfuncreal(A)).(((
  minfuncreal(A)).(q,r)),p) & (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) =(
  minfuncreal(A)).(((minfuncreal(A)).(p,q)),r) & (minfuncreal(A)).(p,((
  minfuncreal(A)).(q,r))) =(minfuncreal(A)).(((minfuncreal(A)).(q,p)),r) & (
minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) =(minfuncreal(A)).(((minfuncreal(A
)).(r,p)),q) & (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) =(minfuncreal(A)).
(((minfuncreal(A)).(r,q)),p) & (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) =(
  minfuncreal(A)).(((minfuncreal(A)).(p,r)),q)
proof
  set s=q"/\"r;
  thus
A1: (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) = s"/\"p by LATTICES:def 2
    .= (minfuncreal(A)).(((minfuncreal(A)).(q,r)),p);
  thus (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) = (minfuncreal(A)).((
  minfuncreal(A)).(p,q),r) by Th11;
  thus (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) = p"/\"s
    .= (q"/\"p)"/\"r by Lm8
    .= (minfuncreal(A)).((minfuncreal(A)).(q,p),r);
  thus
A2: (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) = p"/\"(q"/\"r)
    .= (r"/\"p)"/\"q by Lm8
    .= (minfuncreal(A)).((minfuncreal(A)).(r,p),q);
  thus (minfuncreal(A)).(p,((minfuncreal(A)).(q,r))) =(minfuncreal(A)).(((
  minfuncreal(A)).(r,q)),p) by A1,Th22;
  thus thesis by A2,Th22;
end;
