theorem
  b+c gcd b = 1 & c is odd implies 2*b + c gcd c = 1
  proof
    assume
    A1: b+c gcd b =1 & c is odd; then
    c + b*1 gcd b = 1; then
    A2: c gcd b = 1 by EULER_1:8;
    (2*b + c*1) gcd c = 2*b gcd c by EULER_1:8;
    hence thesis by A1,A2,Lm5a;
  end;
