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

theorem Th24:
  x >= 0 implies sinh(x/2) = sqrt ((cosh(x)-1)/2)
proof
  assume x >= 0;
  then sinh(x/2) = sqrt (sinh(x/2))^2 by SIN_COS5:46,SQUARE_1:22
    .= 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;
