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
  (for A st A is being_line holds f.:A is being_line) implies f is collineation
proof
  assume
A1: for A st A is being_line holds f.:A is being_line;
A2: a<>b implies f.:(Line(a,b))=Line(f.a,f.b)
  proof
    set A=Line(a,b);
    set x=f.a;
    set y=f.b;
    set K=Line(x,y);
    set M=f.:A;
    assume
A3: a<>b;
    then x<>y by FUNCT_2:58;
    then
A4: K is being_line by AFF_1:def 3;
    A is being_line by A3,AFF_1:def 3;
    then
A5: M is being_line by A1;
    a in A by AFF_1:15;
    then
A6: x in M by Th90;
    b in A by AFF_1:15;
    then
A7: y in M by Th90;
    x in K & y in K by AFF_1:15;
    hence thesis by A3,A4,A5,A6,A7,AFF_1:18,FUNCT_2:58;
  end;
A8: f.:A is being_line implies A is being_line
  proof
    set K=f.:A;
    assume f.:A is being_line;
    then consider x,y such that
A9: x<>y and
A10: K=Line(x,y) by AFF_1:def 3;
    y in K by A10,AFF_1:15;
    then consider b such that
    b in A and
A11: f.b=y by Th91;
    x in K by A10,AFF_1:15;
    then consider a such that
    a in A and
A12: f.a=x by Th91;
    set C=Line(a,b);
    C is being_line & f.:C=K by A2,A9,A10,A12,A11,AFF_1:def 3;
    hence thesis by Th92;
  end;
A13: f.:A // f.:C implies A // C
  proof
    set K=f.:A;
    set M=f.:C;
    assume
A14: f.:A // f.:C;
    then M is being_line by AFF_1:36;
    then
A15: C is being_line by A8;
 K is being_line by A14,AFF_1:36;
    then
A16: A is being_line by A8;
    now
      assume
A17:  A<>C;
      assume not A // C;
      then consider p such that
A18:  p in A & p in C by A16,A15,AFF_1:58;
      set x=f.p;
      x in K & x in M by A18,Th90;
      hence contradiction by A14,A17,Th92,AFF_1:45;
    end;
    hence thesis by A16,AFF_1:41;
  end;
A19: f.a,f.b // f.c,f.d implies a,b // c,d
  proof
    set x=f.a;
    set y=f.b;
    set z=f.c;
    set t=f.d;
    assume
A20: f.a,f.b // f.c,f.d;
    now
      set C=Line(c,d);
      set A=Line(a,b);
      set K=f.:A;
      set M=f.:C;
A21:  c in C & d in C by AFF_1:15;
      assume
A22:  a<>b & c <>d;
      then
A23:  x<>y & z<>t by FUNCT_2:58;
      K=Line(x,y) & M=Line(z,t) by A2,A22;
      then
A24:  K // M by A20,A23,AFF_1:37;
      a in A & b in A by AFF_1:15;
      hence thesis by A13,A21,A24,AFF_1:39;
    end;
    hence thesis by AFF_1:3;
  end;
A25: A // C implies f.:A // f.:C
  proof
    assume
A26: A // C;
    then C is being_line by AFF_1:36;
    then
A27: f.:C is being_line by A1;
 A is being_line by A26,AFF_1:36;
    then
A28: f.:A is being_line by A1; then
A29:  f.:A // f.:A by AFF_1:41;
    A<>C implies f.:A // f.:C
    proof
      assume
A30:  A<>C;
      assume not f.:A // f.:C;
      then consider x such that
A31:  x in f.:A and
A32:  x in f.:C by A28,A27,AFF_1:58;
      consider b such that
A33:  b in C and
A34:  x=f.b by A32,Th91;
      consider a such that
A35:  a in A and
A36:  x=f.a by A31,Th91;
      a=b by A36,A34,FUNCT_2:58;
      hence contradiction by A26,A30,A35,A33,AFF_1:45;
    end;
    hence thesis by A29;
  end;
  a,b // c,d implies f.a,f.b // f.c,f.d
  proof
    assume
A37: a,b // c,d;
    now
      set C=Line(c,d);
      set A=Line(a,b);
      set K=f.:A;
      set M=f.:C;
      a in A by AFF_1:15;
      then
A38:  f.a in K by Th90;
      b in A by AFF_1:15;
      then
A39:  f.b in K by Th90;
      d in C by AFF_1:15;
      then
A40:  f.d in M by Th90;
      c in C by AFF_1:15;
      then
A41:  f.c in M by Th90;
      assume a<>b & c <>d;
      then A // C by A37,AFF_1:37;
      hence thesis by A25,A38,A39,A41,A40,AFF_1:39;
    end;
    hence thesis by AFF_1:3;
  end;
  hence thesis by A19,Th87;
end;
