let n be Nat; :: thesis: for L1, L2 being Element of line_of_REAL n st L1 is being_line & L1 = L2 holds
L1 // L2
let L1, L2 be Element of line_of_REAL n; :: thesis: ( L1 is being_line & L1 = L2 implies L1 // L2 )
assume L1 is being_line ; :: thesis: ( not L1 = L2 or L1 // L2 )
then consider x0, x1 being Element of REAL n such that
A1: x0 <> x1 and
A2: L1 = Line (x0,x1) ;
assume A3: L1 = L2 ; :: thesis: L1 // L2
A4: x1 - x0 = 1 * (x1 - x0) by EUCLID_4:3;
x1 - x0 <> 0* n by A1, Th9;
then x1 - x0 // x1 - x0 by A4;
hence L1 // L2 by A3, A2; :: thesis: verum