let X1, X2, X3, X4, X5 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} implies for x being Element of [:X1,X2,X3,X4,X5:] holds
( x `1 = (((x `1 ) `1 ) `1 ) `1 & x `2 = (((x `1 ) `1 ) `1 ) `2 & x `3 = ((x `1 ) `1 ) `2 & x `4 = (x `1 ) `2 & x `5 = x `2 ) )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3,X4,X5:] holds
( x `1 = (((x `1 ) `1 ) `1 ) `1 & x `2 = (((x `1 ) `1 ) `1 ) `2 & x `3 = ((x `1 ) `1 ) `2 & x `4 = (x `1 ) `2 & x `5 = x `2 )
let x be Element of [:X1,X2,X3,X4,X5:]; :: thesis: ( x `1 = (((x `1 ) `1 ) `1 ) `1 & x `2 = (((x `1 ) `1 ) `1 ) `2 & x `3 = ((x `1 ) `1 ) `2 & x `4 = (x `1 ) `2 & x `5 = x `2 )
thus x `1 = [(x `1 ),(x `2 )] `1 by MCART_1:7
.= ([[(x `1 ),(x `2 )],(x `3 )] `1 ) `1 by MCART_1:7
.= ([(x `1 ),(x `2 ),(x `3 )] `1 ) `1 by MCART_1:def 3
.= (([[(x `1 ),(x `2 ),(x `3 )],(x `4 )] `1 ) `1 ) `1 by MCART_1:7
.= (([(x `1 ),(x `2 ),(x `3 ),(x `4 )] `1 ) `1 ) `1 by MCART_1:def 4
.= ((([(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 )] `1 ) `1 ) `1 ) `1 by MCART_1:7
.= (((x `1 ) `1 ) `1 ) `1 by A1, Th19 ; :: thesis: ( x `2 = (((x `1 ) `1 ) `1 ) `2 & x `3 = ((x `1 ) `1 ) `2 & x `4 = (x `1 ) `2 & x `5 = x `2 )
thus x `2 = [(x `1 ),(x `2 )] `2 by MCART_1:7
.= ([[(x `1 ),(x `2 )],(x `3 )] `1 ) `2 by MCART_1:7
.= ([(x `1 ),(x `2 ),(x `3 )] `1 ) `2 by MCART_1:def 3
.= (([[(x `1 ),(x `2 ),(x `3 )],(x `4 )] `1 ) `1 ) `2 by MCART_1:7
.= (([(x `1 ),(x `2 ),(x `3 ),(x `4 )] `1 ) `1 ) `2 by MCART_1:def 4
.= ((([(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 )] `1 ) `1 ) `1 ) `2 by MCART_1:7
.= (((x `1 ) `1 ) `1 ) `2 by A1, Th19 ; :: thesis: ( x `3 = ((x `1 ) `1 ) `2 & x `4 = (x `1 ) `2 & x `5 = x `2 )
thus x `3 = [[(x `1 ),(x `2 )],(x `3 )] `2 by MCART_1:7
.= [(x `1 ),(x `2 ),(x `3 )] `2 by MCART_1:def 3
.= ([[(x `1 ),(x `2 ),(x `3 )],(x `4 )] `1 ) `2 by MCART_1:7
.= ([(x `1 ),(x `2 ),(x `3 ),(x `4 )] `1 ) `2 by MCART_1:def 4
.= (([(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 )] `1 ) `1 ) `2 by MCART_1:7
.= ((x `1 ) `1 ) `2 by A1, Th19 ; :: thesis: ( x `4 = (x `1 ) `2 & x `5 = x `2 )
thus x `4 = [[(x `1 ),(x `2 ),(x `3 )],(x `4 )] `2 by MCART_1:7
.= [(x `1 ),(x `2 ),(x `3 ),(x `4 )] `2 by MCART_1:def 4
.= ([(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 )] `1 ) `2 by MCART_1:7
.= (x `1 ) `2 by A1, Th19 ; :: thesis: x `5 = x `2
thus x `5 = [(x `1 ),(x `2 ),(x `3 ),(x `4 ),(x `5 )] `2 by MCART_1:7
.= x `2 by A1, Th19 ; :: thesis: verum