let X1, X2, X3, X4 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies for x being Element of [:X1,X2,X3,X4:] holds
( x `1 = ((x `1) `1) `1 & x `2 = ((x `1) `1) `2 & x `3 = (x `1) `2 & x `4 = x `2 ) )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3,X4:] holds
( x `1 = ((x `1) `1) `1 & x `2 = ((x `1) `1) `2 & x `3 = (x `1) `2 & x `4 = x `2 )
let x be Element of [:X1,X2,X3,X4:]; :: thesis: ( x `1 = ((x `1) `1) `1 & x `2 = ((x `1) `1) `2 & x `3 = (x `1) `2 & x `4 = x `2 )
thus x `1 = [(x `1),(x `2)] `1 by Def1
.= ([(x `1),(x `2),(x `3)] `1) `1 by Def1
.= (([(x `1),(x `2),(x `3),(x `4)] `1) `1) `1 by Def1
.= ((x `1) `1) `1 by A1, Th60 ; :: thesis: ( x `2 = ((x `1) `1) `2 & x `3 = (x `1) `2 & x `4 = x `2 )
thus x `2 = [(x `1),(x `2)] `2 by Def2
.= ([(x `1),(x `2),(x `3)] `1) `2 by Def1
.= (([(x `1),(x `2),(x `3),(x `4)] `1) `1) `2 by Def1
.= ((x `1) `1) `2 by A1, Th60 ; :: thesis: ( x `3 = (x `1) `2 & x `4 = x `2 )
thus x `3 = [(x `1),(x `2),(x `3)] `2 by Def2
.= ([(x `1),(x `2),(x `3),(x `4)] `1) `2 by Def1
.= (x `1) `2 by A1, Th60 ; :: thesis: x `4 = x `2
thus x `4 = [(x `1),(x `2),(x `3),(x `4)] `2 by Def2
.= x `2 by A1, Th60 ; :: thesis: verum