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

theorem Th53:
  -1 <= a & a <= 1 & -1 <= b & b <= 1 implies a^2*b^2 <= 1
proof
  assume that
A1: -1<=a and
A2: a<=1 and
A3: -1<=b and
A4: b<=1;
A5: 0<=b^2 by XREAL_1:63;
  a^2<=1^2 by A1,A2,Th49;
  then
A6: a^2*b^2 <= 1*b^2 by A5,XREAL_1:64;
  b^2<=1^2 by A3,A4,Th49;
  hence thesis by A6,XXREAL_0:2;
end;
