let P, Q be PLANE of S; :: thesis: ( A on P & L on P & A on Q & L on Q implies P = Q )

assume that

A4: ( A on P & L on P ) and

A5: ( A on Q & L on Q ) ; :: thesis: P = Q

consider P1 being PLANE of S such that

A6: for P2 being PLANE of S holds

( ( A on P2 & L on P2 ) iff P1 = P2 ) by A1, Th26;

P1 = P by A4, A6;

hence P = Q by A5, A6; :: thesis: verum

assume that

A4: ( A on P & L on P ) and

A5: ( A on Q & L on Q ) ; :: thesis: P = Q

consider P1 being PLANE of S such that

A6: for P2 being PLANE of S holds

( ( A on P2 & L on P2 ) iff P1 = P2 ) by A1, Th26;

P1 = P by A4, A6;

hence P = Q by A5, A6; :: thesis: verum