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 Th95:
  f is collineation & A // C implies f.:A // f.:C
proof
  assume that
A1: f is collineation and
A2: A // C;
  consider a,b,c,d such that
A3: a<>b & c <>d and
A4: a,b // c,d and
A5: A=Line(a,b) & C=Line(c,d) by A2,AFF_1:37;
A6: f.a,f.b // f.c,f.d by A1,A4,Th87;
A7: f.a<>f.b & f.c <>f.d by A3,FUNCT_2:58;
  f.:A=Line(f.a,f.b) & f.:C=Line(f.c,f.d) by A1,A5,Th93;
  hence thesis by A7,A6,AFF_1:37;
end;
