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,Lm21;
  then
A5: (1-d)*a > (1-d)*b by A3,Lm13;
A6: (1-d)*a+d*a = a;
  d*a >= d*c by A1,A4,Lm12;
  hence thesis by A5,A6,Lm8;
end;
