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

theorem Th21:
  sinh(y+z+w) = (tanh(y)+tanh(z)+tanh(w)+tanh(y)*tanh(z)*tanh w) *
cosh(y)*cosh(z)*cosh(w) & cosh(y+z+w) = (1+tanh(y)*tanh(z)+tanh(z)*tanh(w)+tanh
(w)*tanh y) * cosh(y)*cosh(z)*cosh(w) & tanh(y+z+w) = (tanh(y)+tanh(z)+tanh(w)+
  tanh(y)*tanh(z)*tanh w) / (1+tanh(z)*tanh(w)+tanh(w)*tanh(y)+tanh(y)*tanh z)
proof
A1: cosh(w) <> 0 by Lm1;
  cosh(y) <> 0 & cosh(z) <> 0 by Lm1;
  then
A2: cosh(y)*cosh(z) <> 0 by XCMPLX_1:6;
A3: cosh(y+z+w) = cosh(y+z)*cosh(w)+sinh(y+z)*sinh(w) by Lm10
    .= (cosh(y)*cosh(z)+sinh(y)*sinh(z))*cosh(w)+sinh(y+z)*sinh(w) by Lm10
    .= cosh(y)*cosh(z)*cosh(w)+sinh(y)*sinh(z)*cosh(w) + (sinh(y)*cosh(z)+
  cosh(y)*sinh(z))*sinh(w) by Lm10
    .= cosh(y)*cosh(z)*cosh(w)+tanh(y)*tanh(z)*(cosh(y)*cosh(z))*cosh(w) + (
  sinh(y)*sinh(w)*cosh(z)+sinh(z)*sinh(w)*cosh(y)) by Lm14
    .= cosh(y)*cosh(z)*cosh(w) + tanh(y)*tanh(z)*(cosh(y)*cosh(z))*cosh(w) +
  (tanh(w)*tanh(y)*(cosh(w)*cosh(y))*cosh(z)+sinh(z)*sinh(w)*cosh(y)) by Lm14
    .= (1+tanh(y)*tanh(z))*(cosh(y)*cosh(z)*cosh(w)) + (tanh(z)*tanh(w)*(
  cosh(z)*cosh(w))*cosh(y) + tanh(w)*tanh(y)*(cosh(y)*cosh(z)*cosh(w))) by Lm14
    .= (1+tanh(y)*tanh(z)+tanh(z)*tanh(w)+tanh(w)*tanh(y)) *cosh(y)*cosh(z)*
  cosh(w);
A4: sinh(y+z+w) = sinh(y+z)*cosh(w)+cosh(y+z)*sinh(w) by Lm10
    .= (sinh(y)*cosh(z)+cosh(y)*sinh(z))*cosh(w)+cosh(y+z)*sinh(w) by Lm10
    .= (sinh(y)*cosh(z)+cosh(y)*sinh(z))*cosh(w) +(cosh(y)*cosh(z)+sinh(y)*
  sinh(z))*sinh(w) by Lm10
    .= (tanh(y)*(cosh(y)*cosh(z))+cosh(y)*sinh(z))*cosh(w) + (cosh(y)*cosh(z
  )+sinh(y)*sinh(z))*sinh(w) by Lm14
    .= (tanh(y)*(cosh(y)*cosh(z))+tanh(z)*(cosh(y)*cosh(z)))*cosh(w) + (cosh
  (y)*cosh(z)+sinh(y)*sinh(z))*sinh(w) by Lm14
    .= (tanh(y)+tanh(z))*(cosh(y)*cosh(z)*cosh(w)) + (cosh(y)*(cosh(z)*sinh(
  w))+sinh(y)*sinh(z)*sinh(w))
    .= (tanh(y)+tanh(z))*(cosh(y)*cosh(z)*cosh(w)) + (cosh(y)*(tanh(w)*(cosh
  (z)*cosh(w)))+sinh(y)*sinh(z)*sinh(w)) by Lm14
    .= (tanh(y)+tanh(z))*(cosh(y)*cosh(z)*cosh(w)) + (tanh(w)*(cosh(y)*cosh(
  z))*cosh(w) + tanh(y)*cosh(y)*sinh(z)*sinh(w)) by Lm14
    .= (tanh(y)+tanh(z))*(cosh(y)*cosh(z)*cosh(w)) + (tanh(w)*(cosh(y)*cosh(
  z))*cosh(w) + tanh(y)*cosh(y)*(tanh(z)*cosh(z))*sinh(w)) by Lm14
    .= (tanh(y)+tanh(z))*(cosh(y)*cosh(z)*cosh(w)) +(tanh(w)*(cosh(y)*cosh(z
  ))*cosh(w) +tanh(y)*tanh(z)*(cosh(y)*cosh(z))*(tanh(w)*cosh(w))) by Lm14
    .= (tanh(y)+tanh(z)+tanh(w)+tanh(y)*tanh(z)*tanh(w)) *cosh(y)*cosh(z)*
  cosh(w);
  then
  tanh(y+z+w) = ((tanh(y)+tanh(z)+tanh(w)+tanh(y)*tanh(z)*tanh(w)) *(cosh(
  y)*cosh(z)*cosh(w))) /((1+tanh(y)*tanh(z)+tanh(z)*tanh(w)+tanh(w)*tanh(y)) *(
  cosh(y)*cosh(z)*cosh(w))) by A3,Th1
    .= (tanh(y)+tanh(z)+tanh(w)+tanh(y)*tanh(z)*tanh(w)) /(1+tanh(y)*tanh(z)
  +tanh(z)*tanh(w)+tanh(w)*tanh(y)) by A1,A2,XCMPLX_1:6,91;
  hence thesis by A4,A3;
end;
