reserve a,b,c,d for Real;
reserve r,s for Real;

theorem
  d <= 1 & a <= b implies (1-d)*a+d*b <= b
proof
  assume that
A1: d<=1 and
A2: a<=b;
  1-d>=0 by A1,Th48;
  then (1-d)*a<=(1-d)*b by A2,Lm12;
  then (1-d)*a+d*b<=(1-d)*b+d*b by Lm6;
  hence thesis;
end;
