let X be set ; :: thesis: for L being List of X

for O being Operation of X holds L WHEREgt (O,0) = L WHERE O

let L be List of X; :: thesis: for O being Operation of X holds L WHEREgt (O,0) = L WHERE O

let O be Operation of X; :: thesis: L WHEREgt (O,0) = L WHERE O

thus L WHEREgt (O,0) c= L WHERE O by Th8; :: according to XBOOLE_0:def 10 :: thesis: L WHERE O c= L WHEREgt (O,0)

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in L WHERE O or z in L WHEREgt (O,0) )

assume A1: z in L WHERE O ; :: thesis: z in L WHEREgt (O,0)

then reconsider z = z as Element of X ;

A2: ( z . O <> {} & z in L ) by A1, Th3;

then 0 in card (z . O) by ORDINAL3:8;

hence z in L WHEREgt (O,0) by A2; :: thesis: verum

for O being Operation of X holds L WHEREgt (O,0) = L WHERE O

let L be List of X; :: thesis: for O being Operation of X holds L WHEREgt (O,0) = L WHERE O

let O be Operation of X; :: thesis: L WHEREgt (O,0) = L WHERE O

thus L WHEREgt (O,0) c= L WHERE O by Th8; :: according to XBOOLE_0:def 10 :: thesis: L WHERE O c= L WHEREgt (O,0)

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in L WHERE O or z in L WHEREgt (O,0) )

assume A1: z in L WHERE O ; :: thesis: z in L WHEREgt (O,0)

then reconsider z = z as Element of X ;

A2: ( z . O <> {} & z in L ) by A1, Th3;

then 0 in card (z . O) by ORDINAL3:8;

hence z in L WHEREgt (O,0) by A2; :: thesis: verum