reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;
reserve L,L0,L1,L2 for Element of line_of_REAL n;

theorem Th70:
  x1 in L & x2 in L & x3 in L & x1 <> x2 implies ex a st x3 - x1 = a*(x2 - x1)
proof
  assume x1 in L & x2 in L & x3 in L & x1 <> x2;
  then x1 in Line(x1,x2) & x3 in Line(x1,x2) by Th64;
  then consider a such that
A1: x3 - x1 = a*(x2 - x1) by Th31;
  take a;
  thus thesis by A1;
end;
