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
  x2 - x1,x3 - x1 are_lindependent2 & y2 in Line(x1,x2) & y3 in Line(x1,
x3) & L1 = Line(x2,x3) & L2 = Line(y2,y3) implies (L1 // L2 iff ex a st a <> 0
  & y2 - x1 = a*(x2 - x1) & y3 - x1 = a*(x3 - x1))
proof
  assume that
A1: x2 - x1,x3 - x1 are_lindependent2 and
A2: y2 in Line(x1,x2) and
A3: y3 in Line(x1,x3) and
A4: L1 = Line(x2,x3) and
A5: L2 = Line(y2,y3);
  thus L1 // L2 implies ex a st a <> 0 & y2 - x1 = a*(x2 - x1) & y3 - x1 = a*(
  x3 - x1)
  proof
    assume
A6: L1 // L2;
    then L1 is being_line by Th66;
    then
A7: x2 <> x3 by A4,Th76;
    L2 is being_line by A6,Th66;
    then
A8: y2 <> y3 by A5,Th76;
A9: y2 in L2 & y3 in L2 by A5,EUCLID_4:9;
    consider t such that
A10: y3 = (1 - t)*x1 + t*x3 by A3;
    x2 in L1 & x3 in L1 by A4,EUCLID_4:9;
    then
A11: y3 - y2 // x3 - x2 by A6,A7,A8,A9,Th77;
    then consider a such that
A12: y3 - y2 = a*(x3 - x2);
    take a;
    consider s such that
A13: y2 = (1 - s)*x1 + s*x2 by A2;
A14: 0*n = y3 - y2 - a*(x3 - x2) by A12,Th9
      .= (1 - t)*x1 + t*x3 - (1 - s)*x1 - s*x2 - a*(x3 - x2) by A13,A10,
RVSUM_1:39
      .= 1 * x1 + -t*x1 + t*x3 - (1 - s)*x1 - s*x2 - a*(x3 - x2) by Th11
      .= 1 * x1 + -t*x1 + t*x3 - (1 * x1 + -s*x1)- s*x2- a*(x3 - x2) by Th11
      .= 1 * x1 + -t*x1 + t*x3 - 1 * x1 - -s*x1 - s*x2- a*(x3 - x2) by
RVSUM_1:39
      .= 1 * x1 + -t*x1 + -1 * x1 + t*x3 + s*x1 - s*x2- a*(x3 - x2) by
RVSUM_1:15
      .= 1 * x1 + -1 * x1 + -t*x1 + t*x3 + s*x1 - s*x2- a*(x3 - x2) by
RVSUM_1:15
      .= 0*n + -t*x1 + t*x3 + s*x1 - s*x2- a*(x3 - x2) by Th2
      .= -t*x1 + t*x3 + s*x1 - s*x2- a*(x3 - x2) by EUCLID_4:1
      .= t*(x3 - x1) + s*x1 + -s*x2 - a*(x3 - x2) by Th12
      .= t*(x3 - x1) - s*x2 + s*x1 - a*(x3 - x2) by RVSUM_1:15
      .= t*(x3 - x1) - (s*x2 - s*x1) - a*(x3 - x2) by Th4
      .= t*(x3 - x1) - s*(x2 - x1) - a*(x3 - x2) by Th12
      .= t*(x3 - x1) - s*(x2 - x1) - a*(x3 - 0*n - x2) by RVSUM_1:32
      .= t*(x3 - x1) - s*(x2 - x1) - a*(x3 - (x1 - x1) - x2) by Th2
      .= t*(x3 - x1) - s*(x2 - x1) - a*(x3 - x1 + x1 + -x2) by Th4
      .= t*(x3 - x1) - s*(x2 - x1) - a*((x3 - x1) + (-x2 + x1)) by RVSUM_1:15
      .= t*(x3 - x1) - s*(x2 - x1) - (a*(x3 - x1) + a*(-x2 + x1)) by EUCLID_4:6
      .= t*(x3 - x1) - s*(x2 - x1) - a*(x3 - x1) - a*(-x2 + x1) by RVSUM_1:39
      .= t*(x3 - x1) - (a*(x3 - x1) + s*(x2 - x1)) - a*(-x2 + x1) by RVSUM_1:39
      .= t*(x3 - x1) - a*(x3 - x1) - s*(x2 - x1) - a*(-x2 + x1) by RVSUM_1:39
      .= (t - a)*(x3 - x1) - s*(x2 - x1) - a*(-x2 + x1) by Th11
      .= (t - a)*(x3 - x1) - s*(x2 - x1) - (a*(-x2) + a*x1) by EUCLID_4:6
      .= (t - a)*(x3 - x1) - s*(x2 - x1) - (-a*x2 + a*x1) by Th3
      .= (t - a)*(x3 - x1) - s*(x2 - x1) - -a*x2 - a*x1 by RVSUM_1:39
      .= (t - a)*(x3 - x1) - s*(x2 - x1) + (a*x2 + - a*x1) by RVSUM_1:15
      .= (t - a)*(x3 - x1) - s*(x2 - x1) + a*(x2 - x1) by Th12
      .= (t - a)*(x3 - x1) - (s*(x2 - x1) - a*(x2 - x1)) by Th4
      .= (t - a)*(x3 - x1) + -(s - a)*(x2 - x1) by Th11
      .= (t - a)*(x3 - x1) + (-(s - a))*(x2 - x1) by Th3
      .= (t - a)*(x3 - x1) + (a - s)*(x2 - x1);
    then t - a = 0 by A1;
    then
A15: y3 - x1 = 1 * x1 + -a*x1 + a*x3 - x1 by A10,Th11
      .= x1 + -a*x1 + a*x3 - x1 by EUCLID_4:3
      .= -a*x1 + a*x3 + x1 - x1 by RVSUM_1:15
      .= -a*x1 + a*x3 + (x1 - x1) by Th5
      .= -a*x1 + a*x3 + 0*n by Th2
      .= -a*x1 + a*x3 by EUCLID_4:1
      .= a*(x3 - x1) by Th12;
    a - s = 0 by A1,A14;
    then
A16: y2 - x1 = 1 * x1 + -a*x1 + a*x2 - x1 by A13,Th11
      .= x1 + -a*x1 + a*x2 - x1 by EUCLID_4:3
      .= -a*x1 + a*x2 + x1 - x1 by RVSUM_1:15
      .= -a*x1 + a*x2 + (x1 - x1) by Th5
      .= -a*x1 + a*x2 + 0*n by Th2
      .= -a*x1 + a*x2 by EUCLID_4:1
      .= a*(x2 - x1) by Th12;
    y3 - y2 <> 0*n by A11;
    hence thesis by A12,A16,A15,EUCLID_4:3;
  end;
  now
    assume ex a st a <> 0 & y2 - x1 = a*(x2 - x1) & y3 - x1 = a*(x3 - x1);
    then consider a such that
A17: a <> 0 and
A18: y2 - x1 = a*(x2 - x1) and
A19: y3 - x1 = a*(x3 - x1);
    take a;
    take x2;
    take x3;
    take y2;
    take y3;
    x2 <> x3 by A1,Th37;
    then
A20: x3 - x2 <> 0*n by Th9;
A21: y3 - y2 = x1 + a*(x3 - x1) - y2 by A19,Th6
      .= a*(x3 - x1) + x1 - (x1 + a*(x2 - x1)) by A18,Th6
      .= a*(x3 - x1) + x1 - x1 - a*(x2 - x1) by RVSUM_1:39
      .= a*(x3 - x1) + (x1 - x1) - a*(x2 - x1) by Th5
      .= a*(x3 - x1) + 0*n - a*(x2 - x1) by Th2
      .= a*(x3 - x1) - a*(x2 - x1) by EUCLID_4:1
      .= a*x3 + -a*x1 - a*(x2 - x1) by Th12
      .= a*x3 + -a*x1 - (-a*x1 + a*x2) by Th12
      .= a*x3 + -a*x1 - -a*x1 - a*x2 by RVSUM_1:39
      .= a*x3 + (-a*x1 - -a*x1) - a*x2 by Th5
      .= a*x3 + 0*n - a*x2 by Th2
      .= a*x3 - a*x2 by EUCLID_4:1
      .= a*(x3 - x2) by Th12;
    then y3 - y2 <> 0*n by A17,A20,EUCLID_4:5;
    then y3 - y2 // x3 - x2 by A21,A20;
    hence L1 // L2 by A4,A5;
  end;
  hence thesis;
end;
