reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  a,b are_coprime & a+b divides a*c + b*d implies a+b divides c-d
  proof
    assume
    A1: a,b are_coprime & a+b divides a*c + b*d;
    set u = a*c; set v = -b*d;
    A1a: a+b divides u - v by A1;
    A2: a+b divides a*(u+0) + b*(v+0) by A1a,Th31;
    consider t such that
    A4: a*a*c - b*b*d = (a+b)*t by A2;
    A6: (a*b)*(c-d) = (a*c-b*d-t)*(a+b) by A4;
    a,a+b are_coprime & b,(a+b) are_coprime by A1,EULER_1:7; then
    a*b,a+b are_coprime by WSIERP_1:6;
    hence thesis by A6,INT_1:def 3,WSIERP_1:29;
  end;
