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

theorem Th27:
  sinh(2*x) = 2*tanh(x)/(1-(tanh x)^2) & sinh(3*x) = sinh(x)*(4*(
cosh x)^2-1) & sinh(3*x) = 3*sinh(x)-2*sinh(x)*(1-cosh(2*x)) & cosh(2*x) = 1+2*
(sinh x)^2 & cosh(2*x) = (cosh x)^2+(sinh x)^2 & cosh(2*x) = (1+(tanh x)^2)/(1-
  (tanh x)^2) & cosh(3*x) = cosh(x)*(4*(sinh x)^2+1) & tanh(3*x) = (3*tanh x+(
  tanh x)|^3)/(1+3*(tanh x)^2)
proof
A1: cosh x <> 0 by Lm1;
A2: sinh(2*x) = 2*sinh(x)*cosh(x) by Th26
    .= 2*sinh(x)*cosh(x)*(cosh(x)/cosh(x)) by A1,XCMPLX_1:88
    .= 2*sinh(x)*cosh(x)*cosh(x)/cosh(x) by XCMPLX_1:74
    .= 2*sinh(x)*(cosh(x)*cosh(x))/cosh(x)
    .= 2*sinh(x)/cosh(x)*(cosh(x)*cosh(x)) by XCMPLX_1:74
    .= 2*(sinh(x)/cosh(x))*(cosh(x)*cosh(x)) by XCMPLX_1:74
    .= 2*tanh(x)*(cosh(x)*cosh(x)) by Th1
    .= 2*tanh(x)/(1/(cosh(x)*cosh x)) by XCMPLX_1:100
    .= 2*tanh(x)/((cosh(x)*cosh(x)-(sinh x)^2)/(cosh(x)*cosh x)) by Lm3
    .= 2*tanh(x) / (cosh(x)*cosh(x)/(cosh(x)*cosh(x)) - sinh(x)*sinh(x)/(
  cosh(x)*cosh x)) by XCMPLX_1:120
    .= 2*tanh(x) / (cosh(x)/cosh(x)*(cosh(x)/cosh(x)) - sinh(x)*sinh(x)/(
  cosh(x)*cosh x)) by XCMPLX_1:76
    .= 2*tanh(x) / (1*(cosh(x)/cosh(x)) - sinh(x)*sinh(x)/(cosh(x)*cosh x))
  by A1,XCMPLX_1:60
    .= 2*tanh(x) / (1-sinh(x)*sinh(x)/(cosh(x)*cosh x)) by A1,XCMPLX_1:60
    .= 2*tanh(x) / (1-sinh(x)/cosh(x)*(sinh(x)/cosh x)) by XCMPLX_1:76
    .= 2*tanh(x) / (1-tanh(x)*(sinh(x)/cosh x)) by Th1
    .= 2*tanh(x) / (1-(tanh x)^2) by Th1;
A3: cosh(3*x) = 4*(cosh x)|^(2+1)-3*cosh x by SIN_COS5:44
    .= 4*((cosh x)|^2*cosh x)-3*cosh x by NEWTON:6
    .= (4*(cosh x)|^(1+1)-3)*cosh x
    .= (4*((cosh x)|^1*cosh x)-3)*cosh x by NEWTON:6
    .= (4*((cosh x)^2-(sinh x)^2+(sinh x)^2)-3)*cosh x 
    .= (4*(1+(sinh x)^2)-3)*cosh x by Lm3
    .= cosh(x)*(4*(sinh x)^2+1);
A4: cosh(2*x) = 2*((cosh x)^2-(sinh x)^2+(sinh x)^2)-1 by Th26
    .= 2*(1+(sinh x)^2)-1 by Lm3
    .= 1+2*(sinh x)^2;
A5: tanh(3*x) = tanh(x+x+x)
    .= (tanh(x)+tanh(x)+tanh(x)+tanh(x)*tanh(x)*tanh x) /(1+tanh(x)*tanh(x)+
  tanh(x)*tanh(x)+tanh(x)*tanh x) by Th21
    .= (3*tanh(x)+(tanh(x))|^1*tanh(x)*tanh x)/(1+tanh(x)*tanh(x) +tanh(x)*
  tanh(x)+tanh(x)*tanh x)
    .= (3*tanh(x)+(tanh(x))|^(1+1)*tanh x)/(1+tanh(x)*tanh x +tanh(x)*tanh(x
  )+tanh(x)*tanh x) by NEWTON:6
    .= (3*tanh(x)+(tanh x)|^(1+1+1))/(1+tanh(x)*tanh x +tanh(x)*tanh(x)+tanh
  (x)*tanh(x)) by NEWTON:6
    .= (3*tanh(x)+(tanh x)|^3)/(1+3*(tanh x)^2);
A6: cosh x <> 0 by Lm1;
A7: cosh(2*x) = 2*(cosh x)^2-1 by Th26
    .= 2*(cosh x)^2-((cosh x)^2-(sinh x)^2) by Lm3
    .= (cosh x)^2+(sinh x)^2;
  then
A8: cosh(2*x) = (1/sech x)^2+(sinh x)^2 by Th2
    .= 1/(sech x)^2+(sinh x)^2*1^2 by XCMPLX_1:76
    .= 1/(sech x)^2+((sinh x)^2/(cosh x)^2)*(cosh x)^2 by A6,XCMPLX_1:6,87
    .= 1/(sech x)^2+(sinh x/cosh x)^2*(cosh x)^2 by XCMPLX_1:76
    .= 1/(sech x)^2+(tanh x)^2*(cosh x)^2 by Th1
    .= 1/(sech x)^2+(tanh x)^2/(1^2/(cosh x)^2) by XCMPLX_1:100
    .= 1/(sech x)^2+(tanh x)^2/(1/cosh x)^2 by XCMPLX_1:76
    .= 1/(sech x)^2+(tanh x)^2/(sech x)^2 by SIN_COS5:def 2
    .= (1+(tanh x)^2)/((sech x)^2+(tanh x)^2-(tanh x)^2) by XCMPLX_1:62
    .= (1+(tanh x)^2)/(1-(tanh x)^2) by SIN_COS5:38;
A9: sinh(3*x) = 4*(sinh(x))|^(2+1)+3*sinh x by SIN_COS5:43
    .= 4*((sinh x)|^2*sinh x)+3*sinh x by NEWTON:6
    .= sinh(x)*(4*(sinh x)|^(1+1)+3)
    .= sinh(x)*(4*((sinh x)|^1*sinh x)+3) by NEWTON:6
    .= sinh(x)*(4*((cosh x)^2-((cosh x)^2-(sinh x)^2))+3)
    .= sinh(x)*(4*((cosh x)^2-1)+3) by Lm3
    .= sinh(x)*(4*(cosh x)^2-1);
  then sinh(3*x) = sinh(x)*(4*((cosh x)^2-(sinh x)^2+(sinh x)^2))-sinh x
    .= sinh(x)*(4*(1+(sinh x)^2))-sinh x by Lm3
    .= (4*sinh(x)+sinh(x)*(2*((cosh x)^2-((cosh x)^2-(sinh x)^2)) +2*(sinh x
  )^2))-sinh x
    .= (4*sinh(x)+sinh(x)*(2*((cosh x)^2-1)+2*(sinh x)^2))-sinh x by Lm3
    .= (4*sinh(x)+sinh(x)*(2*((cosh(x)*cosh x) +(sinh x)^2-1)))-sinh x
    .= (4*sinh(x)+sinh(x)*(2*(cosh(x+x)-1)))-sinh(x) by Lm10
    .= 3*sinh(x)-2*sinh(x)*(1-cosh(2*x));
  hence thesis by A2,A9,A4,A7,A8,A3,A5;
end;
