reserve x,y,t for Real;

theorem
  sinh"(x)-sinh"(y)=sinh"(x*sqrt(1+y^2)-y*sqrt(1+x^2))
proof
  set t=(sqrt(x^2+1))*(sqrt(y^2+1))-x*y;
  set q=sqrt((x*sqrt(1+y^2)-y*sqrt(1+x^2))^2+1);
A1: x^2+1>=0 by Lm6;
  y+0<sqrt(y^2+1) by Lm8;
  then
A2: sqrt(y^2+1)-y>0 by XREAL_1:20;
A3: y^2+1>=0 by Lm6;
  (sqrt(x^2+1)*sqrt(y^2+1)-x*y)>=0 by Lm9;
  then
A4: t=sqrt(((sqrt(x^2+1))*(sqrt(y^2+1))-x*y)^2) by SQUARE_1:22
    .=sqrt(((sqrt(x^2+1))^2*(sqrt(y^2+1))^2)-2*((sqrt(x^2+1))* (sqrt(y^2+1))
  )*(x*y)+(x*y)^2)
    .=sqrt(((x^2+1)*(sqrt(y^2+1))^2)-2*((sqrt(x^2+1))* (sqrt(y^2+1)))*(x*y)+
  (x*y)^2) by A1,SQUARE_1:def 2
    .=sqrt(((x^2+1)*(y^2+1))-2*((sqrt(x^2+1))* (sqrt(y^2+1)))*(x*y)+(x*y)^2)
  by A3,SQUARE_1:def 2
    .=sqrt(2*((x*y)^2)+x^2+y^2+1-2*(x*y)*((sqrt(x^2+1))* (sqrt(y^2+1))));
A5: q=sqrt((x^2*(sqrt(1+y^2))^2)-2*(x*sqrt(1+y^2))*(y*sqrt(1+x^2)) +((y*
  sqrt(1+x^2)))^2+1)
    .=sqrt((x^2*(1+y^2))-2*(x*sqrt(1+y^2))*(y*sqrt(1+x^2)) +((y*sqrt(1+x^2))
  )^2+1) by A3,SQUARE_1:def 2
    .=sqrt((x^2+x^2*y^2)-2*(x*sqrt(1+y^2))*(y*sqrt(1+x^2)) +(y^2*(sqrt(1+x^2
  ))^2)+1)
    .=sqrt((x^2+x^2*y^2)-2*(x*sqrt(1+y^2))*(y*sqrt(1+x^2)) +(y^2*(1+x^2))+1)
  by A1,SQUARE_1:def 2
    .=sqrt(x^2+y^2+2*(x*y)^2+1-2*x*y*sqrt(1+y^2)*sqrt(1+x^2));
  sqrt(x^2+1)+x>0 & sqrt(y^2+1)+y>0 by Th5;
  then sinh"(x)-sinh"(y) =log(number_e,(x+sqrt(x^2+1))/(y+sqrt(y^2+1))) by Lm1,
POWER:54,TAYLOR_1:11
    .=log(number_e,((x+sqrt(x^2+1))*(sqrt(y^2+1)-y))/ ((y+sqrt(y^2+1))*(sqrt
  (y^2+1)-y))) by A2,XCMPLX_1:91
    .=log(number_e,((x+sqrt(x^2+1))*(sqrt(y^2+1)-y))/ ((sqrt(y^2+1))^2-y^2))
    .=log(number_e,((x+sqrt(x^2+1))*(sqrt(y^2+1)-y))/ ((y^2+1)-y^2)) by A3,
SQUARE_1:def 2
    .=log(number_e,x*(sqrt(y^2+1))-y*(sqrt(x^2+1))+(sqrt(x^2+1)) *(sqrt(y^2+
  1))-x*y);
  hence thesis by A4,A5;
end;
