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

theorem
  0 <= d & d <= 1 & 0 <= a & 0 <= b & d*a+(1-d)*b = 0 implies d = 0 & b = 0 or
  d = 1 & a = 0 or a = 0 & b = 0
proof
  assume that
A1: 0<=d and
A2: d<=1 and
A3: a>=0 and
A4: b>=0 and
A5: d*a+(1-d)*b=0;
  d-d<=1-d by A2,Lm7;
  then 1-d=0 or b=0 by A1,A3,A4,A5;
  hence thesis by A5;
end;
