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

theorem LCM2:
  for b be non zero Nat holds
    a lcm (a*n + b) = (a*n/b + 1)*(a lcm b)
  proof
    let b be non zero Nat;
    per cases;
    suppose a is zero;
      hence thesis;
    end;
    suppose
      a is non zero; then
      reconsider a,b as non zero Nat;
  B2: a*(a*n+b) = (a lcm (a*n+b))*(a gcd (a*n+b)) by NAT_D:29
      .= (a lcm (a*n+b))*(a gcd b) by NEWTON02:5;
      a*a*n + a*b = (a gcd b)*((a*a*n)/(a gcd b)+ (a*b)/(a gcd b))
        by XCMPLX_1:114; then
      (a lcm (a*n+b)) = ((a*a*n)/(a gcd b)+ (a*b)/(a gcd b)) by B2,XCMPLX_1:5
      .= (a*n)*a/(a gcd b) + (a lcm b)*(a gcd b)/(a gcd b) by NAT_D:29
      .= (a*n)*a/(a gcd b) + (a lcm b) by XCMPLX_1:89
      .= (a*n)*(a/(a gcd b)) + (a lcm b) by XCMPLX_1:74
      .= (a*n)*((a lcm b)/b) + (a lcm b) by GL
      .= (a*n)/b*(a lcm b) + 1*(a lcm b) by XCMPLX_1:75;
      hence thesis;
    end;
  end;
