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

theorem
  0 <= d & d <= 1 & a < b & a < c implies a < (1-d)*b+d*c
proof
  assume that
A1: 0 <= d and
A2: d <= 1 and
A3: a<b and
A4: a<c;
  per cases;
  suppose
    d=0;
    hence thesis by A3;
  end;
  suppose
    d=1;
    hence thesis by A4;
  end;
  suppose
A5: not(d=0 or d=1);
    then d<1 by A2,XXREAL_0:1;
    then 1-d>0 by Lm21;
    then
A6: (1-d)*a<(1-d)*b by A3,Lm13;
A7: (1-d)*a+d*a=a;
    d*a<d*c by A1,A4,A5,Lm13;
    hence thesis by A6,A7,Lm8;
  end;
end;
