reserve X for set;
reserve CS for non empty CollStr;
reserve a,b,c for Point of CS;
reserve CLSP for CollSp;
reserve a,b,c,d,p,q,r for Point of CLSP;
reserve i,j,k for Element of NAT;
reserve CLSP for proper CollSp;
reserve a,b,c,p,q,r for Point of CLSP;
reserve P,Q for LINE of CLSP;

theorem Th18:
  p<>q & p in P & q in P implies Line(p,q) c= P
proof
  assume that
A1: p<>q and
A2: p in P & q in P;
  let x be object;
  consider a,b such that
  a<>b and
A3: P = Line(a,b) by Def7;
  assume x in Line(p,q);
  then consider r be Point of CLSP such that
A4: r=x and
A5: p,q,r are_collinear;
  a,b,p are_collinear & a,b,q are_collinear by A2,A3,Th11;
  then a,b,r are_collinear by A1,A5,Th9;
  hence thesis by A3,A4;
end;
