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 Th62:
  f is dilatation & f.p=p & q<>p & Mid q,p,f.q & p,x,y are_collinear implies
  x,y // f.y,f.x
proof
  assume that
A1: f is dilatation and
A2: f.p=p and
A3: q<>p & Mid q,p,f.q and
A4: p,x,y are_collinear;
A5: Mid y,p,f.y by A1,A2,A3,Th60;
A6: now
    assume
A7: x=p;
    then y,x // x,f.y by A5,DIRAF:def 3;
    hence thesis by A2,A7,DIRAF:2;
  end;
A8: now
    assume that
A9: x<>p and
    y<>p and
A10: x<>y;
    consider u such that
A11: not p,x,u are_collinear by A9,DIRAF:37;
    consider r such that
A12: x,y '||' u,r and
A13: x,u '||' y,r by DIRAF:26;
A14: not x,y,u are_collinear
    proof
      assume
A15:  x,y,u are_collinear;
      x,y,p are_collinear & x,y,x are_collinear by A4,DIRAF:30,31;
      hence contradiction by A10,A11,A15,DIRAF:32;
    end;
    then
A16: x,y // u,r by A12,A13,PASCH:14;
A17: not p,u,r are_collinear
    proof
A18:  now
        assume u=r;
        then u,x '||' u,y by A13,DIRAF:22;
        then u,x,y are_collinear by DIRAF:def 5;
        hence contradiction by A14,DIRAF:30;
      end;
      x,y,p are_collinear by A4,DIRAF:30;
      then x,y '||' x,p by DIRAF:def 5;
      then x,y '||' p,x by DIRAF:22;
      then
A19:  u,r '||' p,x by A10,A12,DIRAF:23;
A20:  u,r,u are_collinear by DIRAF:31;
      assume p,u,r are_collinear;
      then
A21:  u,r,p are_collinear by DIRAF:30;
      p,x '||' p,y by A4,DIRAF:def 5;
      then u,r '||' p,y by A9,A19,DIRAF:23;
      then
A22:  u,r,y are_collinear by A18,A21,DIRAF:33;
      u,r,x are_collinear by A18,A21,A19,DIRAF:33;
      hence contradiction by A14,A18,A22,A20,DIRAF:32;
    end;
    then
A23: u<>r by DIRAF:31;
    set u9=f.u, r9=f.r, x9=f.x, y9=f.y;
A24: not x9,y9,u9 are_collinear by A1,A14,Th46;
    x9,y9 '||' u9,r9 & x9,u9 '||' y9,r9 by A1,A12,A13,Th45;
    then x9,y9 // u9,r9 by A24,PASCH:14;
    then
A25: r9,u9 // y9,x9 by DIRAF:2;
    u,r // f.r,f.u by A1,A2,A3,A17,Th61;
    then x,y // r9,u9 by A16,A23,DIRAF:3;
    hence thesis by A25,A23,DIRAF:3,FUNCT_2:58;
  end;
  Mid x,p,f.x by A1,A2,A3,Th60;
  hence thesis by A2,A6,A8,DIRAF:4,def 3;
end;
