let n be Nat; :: thesis: for P1, P2 being Element of plane_of_REAL n st P1 is being_plane & P1 c= P2 holds
P1 = P2
let P1, P2 be Element of plane_of_REAL n; :: thesis: ( P1 is being_plane & P1 c= P2 implies P1 = P2 )
assume that
A1: P1 is being_plane and
A2: P1 c= P2 ; :: thesis: P1 = P2
consider x1, x2, x3 being Element of REAL n such that
A3: ( x2 - x1,x3 - x1 are_lindependent2 & P1 = plane (x1,x2,x3) ) by A1;
A4: x3 in plane (x1,x2,x3) by Th82;
( x1 in plane (x1,x2,x3) & x2 in plane (x1,x2,x3) ) by Th82;
hence P1 = P2 by A2, A3, A4, Th92; :: thesis: verum