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;
reserve P,P0,P1,P2 for Element of plane_of_REAL n;

theorem Th97:
  L1 // L2 implies L1,L2 are_coplane
proof
  assume L1 // L2;
  then consider x1, x2, y1, y2 being Element of REAL n such that
A1: L1 = Line(x1,x2) and
A2: L2 = Line(y1,y2) and
A3: (x2 - x1) // (y2 - y1);
  consider a such that
  a <> 0 and
A4: x2 - x1 = a * (y2 - y1) by A3,Th32;
A5: 1 + (-a) + a = 1;
  y1 in plane(x1,y1,y2) & y2 in plane(x1,y1,y2) by Th82;
  then
A6: L2 c= plane(x1,y1,y2) by A2,Th85;
  x2 = x1 + a * (y2 - y1) by A4,Th6
    .= 1 * x1 + a * (y2 - y1) by EUCLID_4:3
    .= 1 * x1 + (a * y2 + (-a) * y1) by Th12
    .= 1 * x1 + (-a) * y1 + a * y2 by RVSUM_1:15;
  then
A7: x2 in plane(x1,y1,y2) by A5;
  x1 in plane(x1,y1,y2) by Th82;
  then L1 c= plane(x1,y1,y2) by A1,A7,Th85;
  hence thesis by A6;
end;
