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;

theorem
  f is collineation & K is being_line implies f.:K is being_line
proof
  assume that
A1: f is collineation and
A2: K is being_line;
  consider a,b such that
A3: a<>b and
A4: K=Line(a,b) by A2,AFF_1:def 3;
  set q=f.b;
  set p=f.a;
A5: p<>q by A3,FUNCT_2:58;
  f.:K=Line(p,q) by A1,A4,Th93;
  hence thesis by A5,AFF_1:def 3;
end;
