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

theorem
  0 <= d & d <= 1 & b <= a & c <= a implies (1-d)*b+d*c <= a
proof
  assume that
A1: 0<=d and
A2: d<=1 and
A3: a>=b and
A4: a>=c;
  1-d>=0 by A2,Th48;
  then
A5: (1-d)*a>=(1-d)*b by A3,Lm12;
A6: (1-d)*a+d*a = a;
  d*a>=d*c by A1,A4,Lm12;
  hence thesis by A5,A6,Lm6;
end;
