reserve x,y,t for Real;

theorem Th26:
  1<=x & 1<=y & |.y.|<=|.x.| implies 0<=y*sqrt(x^2-1)-x*sqrt(y^2 -1)
proof
  assume that
A1: 1<=x and
A2: 1<=y and
A3: |.y.|<=|.x.|;
A4: 0<=y^2-1 by A2,Lm3;
A5: |.x.|^2=x^2 & |.y.|^2=y^2 by COMPLEX1:75;
  1<=|.y.| by A2,ABSVALUE:def 1;
  then y^2<=x^2 by A3,A5,SQUARE_1:15;
  then (-1)*x^2<=(-1)*y^2 by XREAL_1:65;
  then -x^2+x^2*y^2<=-y^2+x^2*y^2 by XREAL_1:6;
  then 0<=x^2 & sqrt(x^2*(y^2-1))<=sqrt(y^2*(x^2-1)) by A1,A4,SQUARE_1:26;
  then sqrt(x^2)*sqrt(y^2-1)<=sqrt(y^2*(x^2-1)) by A4,SQUARE_1:29;
  then
A6: x*sqrt(y^2-1)<=sqrt(y^2*(x^2-1)) by A1,SQUARE_1:22;
  0<=y^2 & 0<=x^2-1 by A1,Lm3,XREAL_1:63;
  then x*sqrt(y^2-1)<=sqrt(y^2)*sqrt(x^2-1) by A6,SQUARE_1:29;
  then x*sqrt(y^2-1)<=y*sqrt(x^2-1) by A2,SQUARE_1:22;
  then x*sqrt(y^2-1)-x*sqrt(y^2-1)<=y*sqrt(x^2-1)-x*sqrt(y^2-1) by XREAL_1:13;
  hence thesis;
end;
