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;

theorem Th87:
  f is collineation iff for x,y,z,t holds (x,y // z,t iff f.x,f.y // f.z,f.t)
proof
  thus f is collineation implies for x,y,z,t holds x,y // z,t iff f.x,f.y // f
  .z,f.t
  proof
    assume
A1: f is_automorphism_of the CONGR of AFS;
    let x,y,z,t;
    thus x,y // z,t implies f.x,f.y // f.z,f.t
    proof
      assume x,y // z,t;
      then [[x,y],[z,t]] in the CONGR of AFS by ANALOAF:def 2;
      then [[f.x,f.y],[f.z,f.t]] in the CONGR of AFS by A1;
      hence thesis by ANALOAF:def 2;
    end;
    assume f.x,f.y // f.z,f.t;
    then [[f.x,f.y],[f.z,f.t]] in the CONGR of AFS by ANALOAF:def 2;
    then [[x,y],[z,t]] in the CONGR of AFS by A1;
    hence thesis by ANALOAF:def 2;
  end;
  assume
A2: for x,y,z,t holds (x,y // z,t iff f.x,f.y // f.z,f.t);
  let x,y,z,t;
  x,y // z,t iff f.x,f.y // f.z,f.t by A2;
  hence thesis by ANALOAF:def 2;
end;
