reserve AP for AffinPlane;
reserve a,a9,b,b9,c,c9,d,x,y,o,p,q for Element of AP;
reserve A,C,D9,M,N,P for Subset of AP;

theorem
  AP is satisfying_DES1 implies AP is satisfying_DES1_1
proof
  assume
A1: AP is satisfying_DES1;
  let A,P,C,o,a,a9,b,b9,c,c9,p,q;
  assume that
A2: A is being_line and
A3: P is being_line and
A4: C is being_line and
A5: P<>A and
A6: P<>C and
A7: A<>C and
A8: o in A and
A9: a in A and
A10: a9 in A and
A11: o in P and
A12: b in P & b9 in P and
A13: o in C & c in C and
A14: c9 in C and
A15: o<>a and
A16: o<>b and
A17: o<>c and
A18: p<>q and
A19: c <>q and
A20: not LIN b,a,c and
A21: not LIN b9,a9,c9 and
A22: LIN b,a,p and
A23: LIN b9,a9,p and
A24: LIN b,c,q and
A25: LIN b9,c9,q and
A26: a,c // p,q;
A27: LIN o,a,a9 & LIN b9,p,a9 by A2,A8,A9,A10,A23,AFF_1:6,21;
  set K=Line(b,a);
A28: a in K by AFF_1:15;
  then
A29: K<>P by A2,A3,A5,A8,A9,A11,A15,AFF_1:18;
A30: not LIN o,a,c
  proof
    assume LIN o,a,c;
    then c in A by A2,A8,A9,A15,AFF_1:25;
    hence contradiction by A2,A4,A7,A8,A13,A17,AFF_1:18;
  end;
A31: p in K by A22,AFF_1:def 2;
A32: LIN o,c,c9 & LIN b9,q,c9 by A4,A13,A14,A25,AFF_1:6,21;
A33: b<>c & a <> p by A19,A20,A24,A26,AFF_1:7,14;
A34: a9<>c9 & b<>a by A20,A21,AFF_1:7;
  b<>a by A20,AFF_1:7;
  then
A35: K is being_line by AFF_1:def 3;
  set M=Line(b,c);
A36: c in M by AFF_1:15;
  then
A37: M<>P by A3,A4,A6,A11,A13,A17,AFF_1:18;
  b<>c by A20,AFF_1:7;
  then
A38: M is being_line by AFF_1:def 3;
A39: b in M & q in M by A24,AFF_1:15,def 2;
  q<>b
  proof
    assume
A40: q=b;
    ( not LIN b,c,a)& c,a // q,p by A20,A26,AFF_1:4,6;
    hence contradiction by A18,A22,A40,AFF_1:55;
  end;
  then
A41: q<>b9 by A3,A12,A38,A39,A37,AFF_1:18;
A42: b in K by AFF_1:15;
  p<>b by A18,A20,A24,A26,AFF_1:55;
  then
A43: p<>b9 by A3,A12,A35,A42,A31,A29,AFF_1:18;
A44: not LIN b9,p,q
  proof
    set N=Line(p,q);
A45: N is being_line by A18,AFF_1:def 3;
    assume LIN b9,p,q;
    then LIN p,q,b9 by AFF_1:6;
    then
A46: b9 in N by AFF_1:def 2;
    q in N & LIN q,b9,c9 by A25,AFF_1:6,15;
    then
A47: c9 in N by A41,A45,A46,AFF_1:25;
    p in N & LIN p,b9,a9 by A23,AFF_1:6,15;
    then a9 in N by A43,A45,A46,AFF_1:25;
    hence contradiction by A21,A45,A46,A47,AFF_1:21;
  end;
  K<>M by A20,A35,A42,A28,A36,AFF_1:21;
  hence
  thesis by A1,A3,A11,A12,A16,A26,A35,A38,A42,A28,A36,A31,A39,A37,A29,A34,A30
,A44,A33,A27,A32;
end;
