reserve a,b,c,d for Real;

theorem
  a <= b & b <= c implies (#)(a,b) = (#)(a,c) & (b,c)(#) = (a,c)(#)
proof
  assume that
A1: a <= b and
A2: b <= c;
  thus (#)(a,b) = a by A1,Def1
    .= (#)(a,c) by A1,A2,Def1,XXREAL_0:2;
  thus (b,c)(#) = c by A2,Def2
    .= (a,c)(#) by A1,A2,Def2,XXREAL_0:2;
end;
