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

theorem
  y <> 0 implies coth(y)+tanh(z) = cosh(y+z)/(sinh(y)*cosh(z)) & coth(y)
  -tanh(z) = cosh(y-z)/(sinh(y)*cosh(z))
proof
  assume
A1: y <> 0;
A2: cosh(z) <> 0 by Lm1;
A3: coth(y)-tanh(z) = cosh(y)/sinh(y)-tanh(z) by SIN_COS5:def 1
    .= cosh(y)*cosh(z)/(sinh(y)*cosh(z))-tanh(z) by A2,XCMPLX_1:91
    .= cosh(y)*cosh(z)/(sinh(y)*cosh(z))-sinh(z)/cosh(z) by Th1
    .= cosh(y)*cosh(z)/(sinh(y)*cosh(z)) -sinh(y)*sinh(z)/(sinh(y)*cosh(z))
  by A1,Lm4,XCMPLX_1:91
    .= (cosh(y)*cosh(z)-sinh(y)*sinh(z))/(sinh(y)*cosh(z)) by XCMPLX_1:120
    .= cosh(y-z)/(sinh(y)*cosh(z)) by Lm10;
  coth(y)+tanh(z) = cosh(y)/sinh(y)+tanh(z) by SIN_COS5:def 1
    .= cosh(y)*cosh(z)/(sinh(y)*cosh z)+tanh z by A2,XCMPLX_1:91
    .= cosh(y)*cosh(z)/(sinh(y)*cosh z)+sinh(z)/cosh(z) by Th1
    .= cosh(y)*cosh(z)/(sinh(y)*cosh z) +sinh(y)*sinh(z)/(sinh(y)*cosh z) by A1
,Lm4,XCMPLX_1:91
    .= (cosh(y)*cosh(z)+sinh(y)*sinh(z))/(sinh(y)*cosh(z)) by XCMPLX_1:62
    .= cosh(y+z)/(sinh(y)*cosh(z)) by Lm10;
  hence thesis by A3;
end;
