let X1, X2, X3, Y1, Y2, Y3 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & Y1 <> {} & Y2 <> {} & Y3 <> {} implies for x being Element of [:X1,X2,X3:]
for y being Element of [:Y1,Y2,Y3:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 ) )
assume that
A1: ( X1 <> {} & X2 <> {} & X3 <> {} ) and
A2: ( Y1 <> {} & Y2 <> {} & Y3 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3:]
for y being Element of [:Y1,Y2,Y3:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 )
let x be Element of [:X1,X2,X3:]; :: thesis: for y being Element of [:Y1,Y2,Y3:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 )
let y be Element of [:Y1,Y2,Y3:]; :: thesis: ( x = y implies ( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 ) )
assume A3: x = y ; :: thesis: ( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 )
thus x `1 = (x `1) `1 by A1, Th50
.= y `1 by A2, A3, Th50 ; :: thesis: ( x `2 = y `2 & x `3 = y `3 )
thus x `2 = (x `1) `2 by A1, Th50
.= y `2 by A2, A3, Th50 ; :: thesis: x `3 = y `3
thus x `3 = x `2 by A1, Th50
.= y `3 by A2, A3, Th50 ; :: thesis: verum