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

theorem Th46:
  sinh_C/.(x+y*<i>) = sinh.x*cos.y+cosh.x*sin.y*<i>
proof
  sinh_C/.(x+y*<i>) = sinh_C/.((y+(-x)*<i>)*<i>)
    .= <i>*sin_C/.(y+(-x)*<i>) by Th17
    .= <i>*(sin.y*cosh.x+(-cos.y*sinh.x)*<i>) by Th43
    .= --sinh.x*cos.y+cosh.x*sin.y*<i>;
  hence thesis;
end;
