reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem Th7:
  a+b+c is odd & a,b,c are_mutually_coprime implies
  a is odd & b is odd & c is odd
  proof
    assume that
A1: a+b+c is odd and
A2: a,b,c are_mutually_coprime;
    assume a is even or b is even or c is even;
    then per cases;
    suppose
A3:   a is even;
      then b is odd & c is even or b is even & c is odd by A1;
      then not a,c are_coprime or not a,b are_coprime by A3;
      hence contradiction by A2;
    end;
    suppose
A4:   b is even;
      then a is odd & c is even or a is even & c is odd by A1;
      then not b,c are_coprime or not b,a are_coprime by A4;
      hence contradiction by A2;
    end;
    suppose
A5:   c is even;
      then a is odd & b is even or a is even & b is odd by A1;
      then not c,b are_coprime or not c,a are_coprime by A5;
      hence contradiction by A2;
    end;
  end;
