reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;
reserve CS for CongrSpace;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;
reserve AFS for AffinSpace;
reserve a,b,c,d,d1,d2,p,x,y,z,t for Element of AFS;
reserve f,g for Permutation of the carrier of AFS;
reserve A,C,K for Subset of AFS;
reserve AFP for AffinPlane,
  A,C,D,K for Subset of AFP,
  a,b,c,d,p,x,y for Element of AFP,
  f for Permutation of the carrier of AFP;

theorem
  f is collineation & K is being_line & (for x st x in K holds f.x=x) &
  not p in K & f.p=p implies f=id the carrier of AFP
proof
  assume that
A1: f is collineation and
A2: K is being_line and
A3: for x st x in K holds f.x=x and
A4: not p in K and
A5: f.p=p;
A6: for x holds f.x=x
  proof
    let x;
    now
      assume not x in K;
      consider a,b such that
A7:   a in K and
A8:   b in K and
A9:   a<>b by A2,AFF_1:19;
      set A=Line(p,a);
A10:  p in A by AFF_1:15;
      f.:A=Line(f.p,f.a) by A1,Th93;
      then
A11:  f.:A=A by A3,A5,A7;
      A is being_line by A4,A7,AFF_1:def 3;
      then consider C such that
A12:  x in C and
A13:  A // C by AFF_1:49;
A14:  C is being_line by A13,AFF_1:36;
      f.:A // f.:C by A1,A13,Th95;
      then
A15:  f.:C // C by A13,A11,AFF_1:44;
A16:  a in A by AFF_1:15;
      not C // K
      proof
        assume C // K;
        then A // K by A13,AFF_1:44;
        hence contradiction by A4,A7,A10,A16,AFF_1:45;
      end;
      then consider c such that
A17:  c in C and
A18:  c in K by A2,A14,AFF_1:58;
      f.c = c by A3,A18;
      then c in f.:C by A17,Th90;
      then
A19:  f.:C=C by A17,A15,AFF_1:45;
      set M=Line(p,b);
A20:  b in M by AFF_1:15;
      f.:M=Line(f.p,f.b) by A1,Th93;
      then
A21:  f.:M=M by A3,A5,A8;
      M is being_line by A4,A8,AFF_1:def 3;
      then consider D such that
A22:  x in D and
A23:  M // D by AFF_1:49;
A24:  D is being_line by A23,AFF_1:36;
      f.:M // f.:D by A1,A23,Th95;
      then
A25:  f.:D // D by A23,A21,AFF_1:44;
A26:  p in M by AFF_1:15;
      not D // K
      proof
        assume D // K;
        then M // K by A23,AFF_1:44;
        hence contradiction by A4,A8,A26,A20,AFF_1:45;
      end;
      then consider d such that
A27:  d in D and
A28:  d in K by A2,A24,AFF_1:58;
      f.d=d by A3,A28;
      then d in f.:D by A27,Th90;
      then
A29:  f.:D=D by A27,A25,AFF_1:45;
A30:  A is being_line by A13,AFF_1:36;
      x=f.x
      proof
        assume
A31:    x<>f.x;
        f.x in C & f.x in D by A12,A22,A19,A29,Th90;
        then C=D by A12,A22,A14,A24,A31,AFF_1:18;
        then A // M by A13,A23,AFF_1:44;
        then A=M by A10,A26,AFF_1:45;
        hence contradiction by A2,A4,A7,A8,A9,A30,A10,A16,A20,AFF_1:18;
      end;
      hence thesis;
    end;
    hence thesis by A3;
  end;
  for x holds f.x=(id the carrier of AFP).x
  by A6;
  hence thesis by FUNCT_2:63;
end;
