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
  a=b implies Line(a,b) = the carrier of CLSP
proof
  assume
A1: a=b;
  for x be object holds x in Line(a,b) iff x in the carrier of CLSP
  proof
    let x be object;
    thus x in Line(a,b) implies x in the carrier of CLSP
    proof
      assume x in Line(a,b);
      then ex p st x=p & a,b,p are_collinear;
      then reconsider x as Point of CLSP;
      x is Element of CLSP;
      hence thesis;
    end;
    thus x in the carrier of CLSP implies x in Line(a,b)
    proof
      assume x in the carrier of CLSP;
      then reconsider x as Point of CLSP;
      a,b,x are_collinear by A1,Th2;
      hence thesis;
    end;
  end;
  hence thesis by TARSKI:2;
end;
