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;

theorem Th31:
  for x1, x2 being Element of REAL n holds y1 in Line(x1,x2) & y2
  in Line(x1,x2) implies ex a st y2 - y1 =a*(x2 - x1)
proof
  let x1, x2 be Element of REAL n;
  assume y1 in Line(x1,x2);
  then consider t such that
A1: y1 = (1-t)*x1 + t*x2;
  assume y2 in Line(x1,x2);
  then consider s such that
A2: y2 = (1-s)*x1 + s*x2;
  take s - t;
  y2 - y1 = (1-s)*x1 + s*x2 - (1-t)*x1 - t*x2 by A1,A2,RVSUM_1:39
    .= s*x2 + (1-s)*x1 +- t*x2 +- (1-t)*x1 by RVSUM_1:15
    .= s*x2 + - t*x2 + (1-s)*x1 +- (1-t)*x1 by RVSUM_1:15
    .= (s - t)*x2 + (1-s)*x1 +- (1-t)*x1 by Th11
    .= (s - t)*x2 + ((1-s)*x1 +- (1-t)*x1) by RVSUM_1:15
    .= (s - t)*x2 + ((1-s)- (1-t))*x1 by Th11
    .= (s - t)*x2 + (-(s - t))*x1
    .= (s - t)*(x2 - x1) by Th12;
  hence thesis;
end;
