reserve x,y,t for Real;

theorem
  (x*y+sqrt(x^2+1)*sqrt(y^2+1))>=0 implies sinh"(x)+sinh"(y)=sinh"(x*
  sqrt(1+y^2)+y*sqrt(1+x^2))
proof
  assume
A1: (x*y+sqrt(x^2+1)*sqrt(y^2+1))>=0;
  sqrt(x^2+1)+x>0 & sqrt(y^2+1)+y>0 by Th5;
  then
A2: sinh"(x)+sinh"(y) =log(number_e,(x+sqrt(x^2+1))*(y+sqrt(y^2+1))) by Lm1,
POWER:53,TAYLOR_1:11
    .=log(number_e,(x*sqrt(y^2+1)+sqrt(x^2+1)*y+(x*y+sqrt(x^2+1)*sqrt(y^2+1)
  )))
    .=log(number_e,(x*sqrt(y^2+1)+sqrt(x^2+1)*y+ sqrt((x*y+sqrt(x^2+1)*sqrt(
  y^2+1))^2))) by A1,SQUARE_1:22;
A3: y^2+1>=0 by Lm6;
  set p=sqrt(((x*sqrt(1+y^2))+(y*sqrt(1+x^2)))^2+1);
  set t=sqrt((x*y+sqrt(x^2+1)*sqrt(y^2+1))^2);
A4: x^2+1>=0 by Lm6;
A5: p=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*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*y)^2+2*(x*sqrt(1+y^2))*(y*sqrt(1+x^2)) +y^2*(1+x^2))+1)
  by A4,SQUARE_1:def 2
    .=sqrt(x^2+2*(x*y)^2+y^2+1+2*x*y*(sqrt(1+y^2)*sqrt(1+x^2)))
    .=sqrt(x^2+2*(x*y)^2+y^2+1+2*x*y*sqrt((1+y^2)*(1+x^2))) by A4,A3,
SQUARE_1:29;
  t=sqrt((x*y+sqrt((x^2+1)*(y^2+1)))^2) by A4,A3,SQUARE_1:29
    .=sqrt((x*y)^2+2*(x*y)*(sqrt((x^2+1)*(y^2+1)))+(sqrt((x^2+1)*(y^2+1)))^2
  )
    .=sqrt((x*y)^2+2*(x*y)*(sqrt((x^2+1)*(y^2+1)))+((x*y)^2+x^2+y^2+1)) by A4
,A3,SQUARE_1:def 2
    .=sqrt(2*(x*y)^2+x^2+y^2+1+2*(x*y)*(sqrt((x^2+1)*(y^2+1))));
  hence thesis by A2,A5;
end;
