theorem Th49:
  cosh_C/.(x+(-y)*<i>) = cosh.x*cos.y+(-sinh.x*sin.y)*<i>
proof
  cosh_C/.(x+(-y)*<i>) = cosh.x*cos.-y+sinh.x*sin.(-y)*<i> by Th48
    .=cosh.x*cos.y+sinh.x*(sin.-y)*<i> by SIN_COS:30
    .=cosh.x*cos.y+sinh.x*(-sin.y)*<i> by SIN_COS:30;
  hence thesis;
end;
