reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;

theorem Th18:
  for x holds (x in [:AfLines(AS),{1}:] iff ex X st x=[X,1] & X is being_line)
proof
  let x;
A1: now
    assume x in [:AfLines(AS),{1}:];
    then consider x1,x2 being object such that
A2: x1 in AfLines(AS) and
A3: x2 in {1} and
A4: x=[x1,x2] by ZFMISC_1:def 2;
    consider X such that
A5: x1=X and
A6: X is being_line by A2;
    take X;
    thus x=[X,1] by A3,A4,A5,TARSKI:def 1;
    thus X is being_line by A6;
  end;
  now
    given X such that
A7: x=[X,1] and
A8: X is being_line;
    X in AfLines(AS ) by A8;
    hence x in [:AfLines(AS),{1}:] by A7,ZFMISC_1:106;
  end;
  hence thesis by A1;
end;
