reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem Th1:
  0 < a implies ex m st 0 < a*m+b
  proof
    assume
A1: 0 < a; then
A2: (-b)/a*a = -b by XCMPLX_1:87;
    consider m such that
A3: -b/a < m by SEQ_4:3;
    take m;
    (-b/a)*a < m*a by A1,A3,XREAL_1:68;
    then -b+b < a*m+b by A2,XREAL_1:8;
    hence thesis;
  end;
