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 Th98:
  L1 is being_line & L2 is being_line & L1,L2 are_coplane & L1
  misses L2 implies ex P st L1 c= P & L2 c= P & P is being_plane
proof
  assume that
A1: L1 is being_line and
A2: L2 is being_line and
A3: L1,L2 are_coplane;
  consider x1,x2,x3 being Element of REAL n such that
A4: L1 c= plane(x1,x2,x3) & L2 c= plane(x1,x2,x3) by A3;
  consider y2,y3 such that
  y2 <> y3 and
A5: L2 = Line(y2,y3) by A2;
  consider y0,y1 such that
A6: y0 <> y1 and
A7: L1 = Line(y0,y1) by A1;
A8: y0 - y1 <> 0*n by A6,Th9;
  set P = plane(x1,x2,x3);
A9: y2 in L2 by A5,EUCLID_4:9;
  consider y being Element of REAL n such that
A10: y in Line(y0,y1) and
A11: y0 - y1,y2 - y are_orthogonal by Th43;
  assume L1 misses L2;
  then
A12: y <> y2 by A7,A9,A10,XBOOLE_0:3;
  then y2 - y <> 0*n by Th9;
  then
A13: y0 - y1 _|_ y2 - y by A11,A8;
  consider y9 being Element of REAL n such that
A14: y <> y9 and
A15: y9 in L1 by A1,EUCLID_4:14;
  take P;
  y in Line(y,y2) & y2 in Line(y,y2) by EUCLID_4:9;
  then
A16: P in plane_of_REAL n & y9 - y, y2 - y are_lindependent2 by A7,A10,A12,A13
,A14,A15,Th40,Th45;
  then P = plane(y,y9,y2) by A4,A7,A9,A10,A15,Th92;
  hence thesis by A4,A16;
end;
