reserve x,y,t for Real;

theorem Th70:
  (cosh.x)^2=1+(sinh.x)^2
proof
  cosh.(2*x)=cosh.(x+x) .=(cosh.x)^2+(sinh.x)^2 by SIN_COS2:20;
  then 1+2*(sinh.x)^2-(sinh.x)^2=(cosh.x)^2+(sinh.x)^2-(sinh.x)^2 by Th69;
  hence thesis;
end;
