reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem
  exp(-z) = cosh_C/.z - sinh_C/.z
proof
  cosh_C/.z - sinh_C/.z = (exp(z)+exp(-z))/2 - sinh_C/.z by Def4
    .= (exp(z)+exp(-z))/2 - (exp(z)-exp(-z))/2 by Def3
    .= (exp(-z)+exp(-z))/2
    .= (Re(exp(-z))+Re(exp(-z))+(Im(exp(-z))+Im(exp(-z)))*<i>)/2 by
COMPLEX1:81
    .=((2*Re(exp(-z))+2*Im(exp(-z))*<i>))/2
    .=((Re(2*exp(-z))+2*Im(exp(-z))*<i>))/2 by COMSEQ_3:17
    .=((Re(2*exp(-z))+Im(2*exp(-z))*<i>))/2 by COMSEQ_3:17
    .=(2*exp(-z))/2 by COMPLEX1:13
    .=exp(-z)*1;
  hence thesis;
end;
