reserve x,x1,x2,x3 for Real;

theorem
  sinh(x1)<>0 & sinh(x2)<>0 implies coth(x1-x2)=(1-coth(x1)*coth(x2))/(
  coth(x1)-coth(x2))
proof
  assume that
A1: sinh(x1)<>0 and
A2: sinh(x2)<>0;
  -sinh.(x2)<>0 by A2,SIN_COS2:def 2;
  then sinh.(-x2)<>0 by SIN_COS2:19;
  then
A3: sinh(-x2)<>0 by SIN_COS2:def 2;
  coth(x1-x2)=coth(x1+(-x2))
    .=(1+coth(x1)*coth(-x2))/(coth(x1)+coth(-x2)) by A1,A3,Th40
    .=(1+coth(x1)*(-coth(x2)))/(coth(x1)+coth(-x2)) by Lm1
    .=(1-coth(x1)*(coth(x2)))/(coth(x1)+-coth(x2)) by Lm1
    .=(1-coth(x1)*(coth(x2)))/(coth(x1)-coth(x2));
  hence thesis;
end;
