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 Th33:
  x1 // x2 & x2 // x3 implies x1 // x3
proof
  assume that
A1: x1 // x2 and
A2: x2 // x3;
A3: ex t st x1 = t*x3
  proof
    consider b such that
A4: x2 = b*x3 by A2;
    consider a such that
A5: x1 = a*x2 by A1;
    x1 = (a*b)*x3 by A5,A4,EUCLID_4:4;
    hence thesis;
  end;
  x1 <> 0*n & x3 <> 0*n by A1,A2;
  hence thesis by A3;
end;
