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 Th74:
  for x0,x1,y0,y1,L1,L2 st x0 in L1 & x1 in L1 & x0 <> x1 & y0 in
  L2 & y1 in L2 & y0 <> y1 & L1 _|_ L2 holds x1 - x0 _|_ y1 - y0
proof
  let x0,x1,y0,y1,L1,L2;
  assume that
A1: x0 in L1 & x1 in L1 and
A2: x0 <> x1 and
A3: y0 in L2 & y1 in L2 and
A4: y0 <> y1 and
A5: L1 _|_ L2;
  consider x2,x3,y2,y3 such that
A6: L1 = Line(x2,x3) and
A7: L2 = Line(y2,y3) and
A8: x3 - x2 _|_ y3 - y2 by A5;
  consider s such that
A9: x1 - x0 = s*(x3 - x2) by A1,A6,Th31;
  x3 - x2,y3 - y2 are_orthogonal by A8;
  then
A10: |(x3 - x2,y3 - y2)| = 0 by RVSUM_1:def 17;
  consider t such that
A11: y1 - y0 = t*(y3 - y2) by A3,A7,Th31;
  |(x1 - x0,y1 - y0)| = s * |(x3 - x2,y1 - y0)| by A9,EUCLID_4:21
    .= s * (t * |(x3 - x2,y3 - y2)|) by A11,EUCLID_4:22
    .= 0 by A10;
  then
A12: x1 - x0,y1 - y0 are_orthogonal by RVSUM_1:def 17;
  x1 - x0 <> 0*n & y1 - y0 <> 0*n by A2,A4,Th9;
  hence thesis by A12;
end;
