
theorem
  for s1, s3, s4, l being Real st s1 <= s3 & s1 < s4 & 0 <= l & l
  <= 1 holds s1 <= (1-l) * s3+l*s4
proof
  let s1, s3, s4, l be Real;
  assume that
A1: s1 <= s3 and
A2: s1 < s4 and
A3: 0 <= l and
A4: l <= 1;
  now
    per cases;
    suppose
      l=0;
      hence thesis by A1;
    end;
    suppose
      l=1;
      hence thesis by A2;
    end;
    suppose
A5:   not(l=0 or l=1);
      then l<1 by A4,XXREAL_0:1;
      then 1-l>0 by XREAL_1:50;
      then
A6:   (1-l)*s1<=(1-l)*s3 by A1,XREAL_1:64;
A7:   (1-l)*s1+l*s1=1 * s1;
      l*s1<l*s4 by A2,A3,A5,XREAL_1:68;
      hence thesis by A6,A7,XREAL_1:8;
    end;
  end;
  hence thesis;
end;
