reserve a,b,c,x,y,z for Real;

theorem
  -1 <= a & a <= 1 & -1 <= b & b <= 1 implies b*sqrt(1+a^2) <= sqrt(1+b^2)
proof
  assume that
A1: -1<=a and
A2: a<=1 and
A3: -1<=b and
A4: b<=1;
A5: -1<=-b by A4,XREAL_1:24;
  --1>=-b by A3,XREAL_1:24;
  then (--b)*sqrt(1+a^2)<=sqrt(1+(-b)^2) by A1,A2,A5,Th55;
  hence thesis;
end;
