reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem Th54:
  not LIN o,a,b & LIN o,b,b9 & a,b // a9,b9 & a9=o implies b9=o
proof
  assume that
A1: not LIN o,a,b and
A2: LIN o,b,b9 and
A3: a,b // a9,b9 and
A4: a9=o;
A5: now
    assume a,b // a9,b;
    then b,a // b,a9 by Th3;
    then LIN b,a,a9;
    hence contradiction by A1,A4,Th5;
  end;
  a9,b // a9,b9 by A2,A4;
  hence thesis by A3,A4,A5,Th4;
end;
