reserve A for non trivial Nat,
        B,C,n,m,k for Nat,
        e for Nat;
reserve a for non trivial Nat;

theorem Th22:
  for a,b,A,B be Integer st a,b are_coprime holds
    a divides A & b divides B iff a*b divides a*B + b*A
proof
  let a,b,A,B be Integer such that
A1: a,b are_coprime;
  thus a divides A & b divides B implies a*b divides a*B + b*A
  proof
    assume
A2:   a divides A & b divides B;
    then consider x be Integer such that
A3:   A= a*x by INT_1:def 3;
    consider y be Integer such that
A4:   B= b*y by A2,INT_1:def 3;
    a*B +b*A = (a*b)*(y+x) by A3,A4;
    hence thesis by INT_1:def 3;
  end;
  assume a*b divides a*B + b*A;
  then consider x be Integer  such that
A5: a*B+b*A = (a*b)*x by INT_1:def 3;
  a*(-B+b*x) = b*A by A5;
  hence a divides A by A1,INT_2:25,INT_1:def 3;
  b*(-A+a*x) = a*B by A5;
  hence b divides B by A1,INT_2:25,INT_1:def 3;
end;
