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

theorem Th7:
  exp_R(x) = cosh(x)+sinh(x) & exp_R(-x) = cosh(x)-sinh(x) & exp_R(
x) = (cosh(x/2)+sinh(x/2))/(cosh(x/2)-sinh(x/2)) & exp_R(-x) = (cosh(x/2)-sinh(
x/2))/(cosh(x/2)+sinh(x/2)) & exp_R(x) = (1+tanh(x/2))/(1-tanh(x/2)) & exp_R(-x
  ) = (1-tanh(x/2))/(1+tanh(x/2))
proof
A1: exp_R(x/2)>0 by SIN_COS:55;
A2: cosh(x/2) <> 0 by Lm1;
A3: exp_R(-x) = exp_R(-x/2+-x/2) .= exp_R(-x/2)*exp_R(-x/2) by SIN_COS:50
    .= exp_R(-x/2)*exp_R(x/2)*(exp_R(-x/2)/exp_R(x/2)) by A1,XCMPLX_1:90
    .= exp_R(-x/2+x/2)*(exp_R(-x/2)/exp_R(x/2)) by SIN_COS:50
    .= ((exp_R(x/2)+exp_R(-x/2))/2-(exp_R(x/2)-exp_R(-x/2))/2)/exp_R(x/2) by
SIN_COS:51
    .= ((exp_R(x/2)+exp_R(-x/2))/2-sinh(x/2))/exp_R(x/2) by Lm2
    .= (cosh(x/2)-sinh(x/2))/((exp_R(x/2)+exp_R(-x/2))/2 +(exp_R(x/2)/2-
  exp_R(-x/2)/2)) by Lm2
    .= (cosh(x/2)-sinh(x/2))/(cosh(x/2)+(exp_R(x/2)-exp_R(-x/2))/2) by Lm2
    .= (cosh(x/2)-sinh(x/2))/(cosh(x/2)+sinh(x/2)) by Lm2;
  then
A4: exp_R(-x) = (cosh(x/2)-sinh(x/2))/cosh(x/2) /((cosh(x/2)+sinh(x/2))/cosh
  (x/2)) by A2,XCMPLX_1:55
    .= (cosh(x/2)/cosh(x/2)-sinh(x/2)/cosh(x/2)) /((cosh(x/2)+sinh(x/2))/
  cosh(x/2)) by XCMPLX_1:120
    .= (1-sinh(x/2)/cosh(x/2))/((cosh(x/2)+sinh(x/2))/cosh(x/2)) by A2,
XCMPLX_1:60
    .= (1-tanh(x/2))/((cosh(x/2)+sinh(x/2))/cosh(x/2)) by Th1
    .= (1-tanh(x/2))/(cosh(x/2)/cosh(x/2)+sinh(x/2)/cosh(x/2)) by XCMPLX_1:62
    .= (1-tanh(x/2))/(1+sinh(x/2)/cosh(x/2)) by A2,XCMPLX_1:60
    .= (1-tanh(x/2))/(1+tanh(x/2)) by Th1;
A5: exp_R(-x) = (exp_R(x)+exp_R(-x))/2-(exp_R(x)-exp_R(-x))/2
    .= cosh(x)-(exp_R(x)-exp_R(-x))/2 by Lm2
    .= cosh(x)-sinh(x) by Lm2;
A6: exp_R(x) = (exp_R(x)+exp_R(-x))/2+(exp_R(x)-exp_R(-x))/2
    .= cosh(x)+(exp_R(x)-exp_R(-x))/2 by Lm2
    .= cosh(x)+sinh(x) by Lm2;
A7: exp_R(-x/2)>0 by SIN_COS:55;
A8: exp_R(x) = exp_R(x/2+x/2) .= exp_R(x/2)*exp_R(x/2) by SIN_COS:50
    .= exp_R(x/2)*exp_R(-x/2)*(exp_R(x/2)/exp_R(-x/2)) by A7,XCMPLX_1:90
    .= exp_R(x/2+-x/2)*(exp_R(x/2)/exp_R(-x/2)) by SIN_COS:50
    .= ((exp_R(x/2)+exp_R(-x/2))/2+ (exp_R(x/2)-exp_R(-x/2))/2)/exp_R(-x/2)
  by SIN_COS:51
    .= (cosh(x/2)+(exp_R(x/2)-exp_R(-x/2))/2)/exp_R(-x/2) by Lm2
    .= (cosh(x/2)+sinh(x/2))/((exp_R(-x/2)+exp_R(x/2))/2 -(exp_R(x/2)-exp_R(
  -x/2))/2) by Lm2
    .= (cosh(x/2)+sinh(x/2))/((exp_R(x/2)+exp_R(-x/2))/2-sinh(x/2)) by Lm2
    .= (cosh(x/2)+sinh(x/2))/(cosh(x/2)-sinh(x/2)) by Lm2;
  then
  exp_R(x) = (cosh(x/2)+sinh(x/2))/cosh(x/2) /((cosh(x/2)-sinh(x/2))/cosh(
  x/2)) by A2,XCMPLX_1:55
    .= (cosh(x/2)/cosh(x/2)+sinh(x/2)/cosh(x/2)) /((cosh(x/2)-sinh(x/2))/
  cosh(x/2)) by XCMPLX_1:62
    .= (1+sinh(x/2)/cosh(x/2))/((cosh(x/2)-sinh(x/2))/cosh(x/2)) by A2,
XCMPLX_1:60
    .= (1+tanh(x/2))/((cosh(x/2)-sinh(x/2))/cosh(x/2)) by Th1
    .= (1+tanh(x/2))/(cosh(x/2)/cosh(x/2)-sinh(x/2)/cosh(x/2)) by XCMPLX_1:120
    .= (1+tanh(x/2))/(1-sinh(x/2)/cosh(x/2)) by A2,XCMPLX_1:60
    .= (1+tanh(x/2))/(1-tanh(x/2)) by Th1;
  hence thesis by A6,A5,A8,A3,A4;
end;
