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 Th44:
  A // C & p in A & p in C implies A=C
proof
  assume that
A1: A // C and
A2: p in A and
A3: p in C;
A4: for A,C,p holds A // C & p in A & p in C implies A c= C
  proof
    let A,C,p;
    assume that
A5: A // C and
A6: p in A and
A7: p in C;
A8: C is being_line by A5,Th35;
A9: A is being_line by A5;
      let x be object;
      assume
A10:  x in A;
      then reconsider x9=x as Element of AS;
      now
        consider q such that
A11:    p<>q and
A12:    q in C by A8,Th19;
        assume
A13:    x9<>p;
        then A=Line(p,x9) by A6,A9,A10,Lm6;
        then p,x9 // C by A5,A13,Th28,Th42;
        then p,x9 // p,q by A7,A8,A11,A12,Th26;
        then p,q // p,x9 by Th3;
        then
A14:    LIN p,q,x9;
        C=Line(p,q) by A7,A8,A11,A12,Lm6;
        hence x in C by A14,Def2;
      end;
      hence x in C by A7;
  end;
  then
A15: C c= A by A1,A2,A3;
  A c= C by A1,A2,A3,A4;
  hence thesis by A15,XBOOLE_0:def 10;
end;
