reserve AFP for AffinPlane;
reserve a,a9,b,b9,c,c9,d,d9,o,p,p9,q,q9,r,s,t,x,y,z for Element of AFP;
reserve A,A9,C,D,P,B9,M,N,K for Subset of AFP;
reserve f for Permutation of the carrier of AFP;

theorem Th8:
  a,b // K & not a in K & AFP is Moufangian implies ex f st f is_Sc K & f.a=b
proof
  assume that
A1: a,b // K and
A2: not a in K and
A3: AFP is Moufangian;
  consider f such that
A4: for x,y holds (f.x=y iff ( x in K & x=y or (not x in K & ex p,p9 st
  p in K & p9 in K & p,a // p9,x & p,b // p9,y & x,y // K))) by A1,A2,A3,Lm24;
A5: f.a=b by A1,A2,A4,Lm25;
  f is_Sc K by A1,A2,A4,Lm28;
  hence thesis by A5;
end;
