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
  x1 in L & x2 in L implies ex x3 st x3 in L & x3 - x1 = a*(x2 - x1)
proof
  set x3 = (1-a)*x1 + a*x2;
  assume x1 in L & x2 in L;
  then
A1: Line(x1,x2) c= L by Th48;
  x3 = 1 * x1 + -a*x1 + a*x2 by Th11
    .= x1 + -a*x1 + a*x2 by EUCLID_4:3
    .= x1 + (a*x2 + -a*x1) by RVSUM_1:15
    .= x1 + a*(x2 - x1) by Th12;
  then x3 in Line(x1,x2) & x3 - x1 = a*(x2 - x1) by Th6;
  hence thesis by A1;
end;
