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;

theorem Th63:
  f is dilatation & f.p=p & q<>p & Mid q,p,f.q implies f is negative_dilatation
proof
  assume
A1: f is dilatation & f.p=p & q<>p & Mid q,p,f.q;
  x,y // f.y,f.x
  proof
    not p,x,y are_collinear implies x,y // f.y,f.x by A1,Th61;
    hence thesis by A1,Th62;
  end;
  hence thesis;
end;
