reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem Th20:
  LIN a,b,c iff ex A st A is being_line & a in A & b in A & c in A
proof
A1: LIN a,b,c implies ex A st A is being_line & a in A & b in A & c in A
  proof
    assume
A2: LIN a,b,c;
A3: now
      set A=Line(a,b);
A4:   a in A by Th14;
A5:   b in A by Th14;
      assume a<>b;
      then
A6:   A is being_line;
      c in A by A2,Def2;
      hence thesis by A6,A4,A5;
    end;
A7: now
      set A=Line(a,c);
A8:   c in A by Th14;
      assume a<>c;
      then
A9:   A is being_line;
      LIN a,c,b by A2,Th5;
      then
A10:  b in A by Def2;
      a in A by Th14;
      hence thesis by A9,A10,A8;
    end;
    now
      consider x such that
A11:  a<>x by SUBSET_1:50;
      set A=Line(a,x);
A12:  a in A by Th14;
      assume that
A13:  a=b and
A14:  a=c;
      A is being_line by A11;
      hence thesis by A13,A14,A12;
    end;
    hence thesis by A3,A7;
  end;
  (ex A st A is being_line & a in A & b in A & c in A) implies LIN a,b,c
  proof
    given A such that
A15: A is being_line and
A16: a in A and
A17: b in A and
A18: c in A;
    consider p,q such that
A19: p<>q and
A20: A=Line(p,q) by A15;
A21: LIN p,q,b by A17,A20,Def2;
A22: LIN p,q,c by A18,A20,Def2;
    LIN p,q,a by A16,A20,Def2;
    hence thesis by A19,A21,A22,Th7;
  end;
  hence thesis by A1;
end;
