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 Th64:
  f is dilatation & f.p=p & (for x holds p,x // p,f.x) implies for
  y,z holds y,z // f.y,f.z
proof
  assume that
A1: f is dilatation and
A2: f.p=p and
A3: p,x // p,f.x;
A4: not p,y,z are_collinear implies y,z // f.y,f.z
  proof
    assume
A5: not p,y,z are_collinear;
A6: p,y // p,f.y & p,z // p,f.z by A3;
    y,z '||' f.y,f.z by A1,Th34;
    hence thesis by A5,A6,PASCH:13;
  end;
  let y,z;
  p,y,z are_collinear implies y,z // f.y,f.z
  proof
    assume
A7:  p,y,z are_collinear;
A8: now
      assume that
A9:   p<>y and
A10:  y<>z and
      z<>p;
      consider q such that
A11:  not p,y,q are_collinear by A9,DIRAF:37;
A12:  not y,z,q are_collinear
      proof
        assume
A13:    y,z,q are_collinear;
        y,z,p are_collinear & y,z,y are_collinear by A7,DIRAF:30,31;
        hence contradiction by A10,A11,A13,DIRAF:32;
      end;
      then
A14:  not f.y,f.z,f.q are_collinear by A1,Th46;
      consider r such that
A15:  y,z '||' q,r and
A16:  y,q '||' z,r by DIRAF:26;
      f.y,f.z '||' f.q,f.r & f.y,f.q '||' f.z,f.r by A1,A15,A16,Th45;
      then f.y,f.z // f.q,f.r by A14,PASCH:14;
      then
A17:  f.q,f.r // f.y,f.z by DIRAF:2;
A18:  q<>r
      proof
        assume q=r;
        then q,y '||' q,z by A16,DIRAF:22;
        then q,y,z are_collinear by DIRAF:def 5;
        hence contradiction by A12,DIRAF:30;
      end;
      not p,q,r are_collinear
      proof
A19:    q,r,q are_collinear by DIRAF:31;
        assume p,q,r are_collinear;
        then
A20:    q,r,p are_collinear by DIRAF:30;
        y,p,z are_collinear by A7,DIRAF:30;
        then y,p '||' y,z by DIRAF:def 5;
        then y,z '||' p,y by DIRAF:22;
        then q,r '||' p,y by A10,A15,DIRAF:23;
        then q,r,y are_collinear by A18,A20,DIRAF:33;
        hence contradiction by A11,A18,A20,A19,DIRAF:32;
      end;
      then
A21:  q,r // f.q,f.r by A4;
      y,z // q,r by A15,A16,A12,PASCH:14;
      then y,z // f.q,f.r by A18,A21,DIRAF:3;
      hence thesis by A18,A17,DIRAF:3,FUNCT_2:58;
    end;
    now
      assume p=z;
      then z,y // f.z,f.y by A2,A3;
      hence thesis by DIRAF:2;
    end;
    hence thesis by A2,A3,A8,DIRAF:4;
  end;
  hence thesis by A4;
end;
