let X1, X2, X3, X4, X5, Y1, Y2, Y3, Y4, Y5 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {} implies for x being Element of [:X1,X2,X3,X4,X5:]
for y being Element of [:Y1,Y2,Y3,Y4,Y5:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 ) )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} ) and
A2: ( Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3,X4,X5:]
for y being Element of [:Y1,Y2,Y3,Y4,Y5:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 )
let x be Element of [:X1,X2,X3,X4,X5:]; :: thesis: for y being Element of [:Y1,Y2,Y3,Y4,Y5:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 )
let y be Element of [:Y1,Y2,Y3,Y4,Y5:]; :: thesis: ( x = y implies ( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 ) )
assume A3: x = y ; :: thesis: ( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 )
thus x `1 = (((x `1 ) `1 ) `1 ) `1 by A1, Th20
.= y `1 by A2, A3, Th20 ; :: thesis: ( x `2 = y `2 & x `3 = y `3 & x `4 = y `4 & x `5 = y `5 )
thus x `2 = (((x `1 ) `1 ) `1 ) `2 by A1, Th20
.= y `2 by A2, A3, Th20 ; :: thesis: ( x `3 = y `3 & x `4 = y `4 & x `5 = y `5 )
thus x `3 = ((x `1 ) `1 ) `2 by A1, Th20
.= y `3 by A2, A3, Th20 ; :: thesis: ( x `4 = y `4 & x `5 = y `5 )
thus x `4 = (x `1 ) `2 by A1, Th20
.= y `4 by A2, A3, Th20 ; :: thesis: x `5 = y `5
thus x `5 = x `2 by A1, Th20
.= y `5 by A2, A3, Th20 ; :: thesis: verum