reserve x,y for Real;
reserve a,b,c for Element of Real_Lattice;
reserve p,q,r for Element of Real_Lattice;

theorem Th4:
  minreal.(p,(minreal.(q,r)))=minreal.((minreal.(q,r)),p) &
minreal.(p,(minreal.(q,r)))=minreal.((minreal.(p,q)),r) & minreal.(p,(minreal.(
  q,r)))=minreal.((minreal.(q,p)),r) & minreal.(p,(minreal.(q,r)))=minreal.((
  minreal.(r,p)),q) & minreal.(p,(minreal.(q,r)))=minreal.((minreal.(r,q)),p) &
  minreal.(p,(minreal.(q,r)))=minreal.((minreal.(p,r)),q)
proof
  set s=q"/\"r;
  thus
A1: minreal.(p,(minreal.(q,r))) = s"/\"p by LATTICES:def 2
    .= minreal.((minreal.(q,r)),p);
  thus minreal.(p,(minreal.(q,r))) = p"/\"s .= (p"/\"q)"/\"r by XXREAL_0:33
    .= minreal.(minreal.(p,q),r);
  thus minreal.(p,(minreal.(q,r))) = p"/\"s .= (q"/\"p)"/\"r by XXREAL_0:33
    .= minreal.(minreal.(q,p),r);
  thus
A2: minreal.(p,(minreal.(q,r))) = p"/\"(q"/\"r) .= (r"/\"p)"/\"q by XXREAL_0:33
    .= minreal.(minreal.(r,p),q);
  thus minreal.(p,(minreal.(q,r)))=minreal.((minreal.(r,q)),p) by A1,Th2;
  thus thesis by A2,Th2;
end;
