reserve x, y, z, w for Real;
reserve n for Element of NAT;

theorem Th25:
  x < 0 implies sinh(x/2) = - sqrt ((cosh(x)-1)/2)
proof
  assume x < 0;
  then
A1: x/2 < 0 by XREAL_1:141;
  sinh(x/2) = --sinh(x/2) .= - sqrt (sinh(x/2))^2 by A1,SIN_COS5:47,SQUARE_1:23
    .= - sqrt (sinh.(x/2))^2 by SIN_COS2:def 2
    .= - sqrt (1/2*(cosh.(2*(x/2))-1)) by SIN_COS2:18
    .= - sqrt ((cosh.x-1)/2)
    .= - sqrt ((cosh(x)-1)/2) by SIN_COS2:def 4;
  hence thesis;
end;
