let X1, X2, X3, X4, X5, X6, X7, X8, X9 be set ; :: thesis: for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A1 be Subset of X1; :: thesis: for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A2 be Subset of X2; :: thesis: for A3 being Subset of X3
for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A3 be Subset of X3; :: thesis: for A4 being Subset of X4
for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A4 be Subset of X4; :: thesis: for A5 being Subset of X5
for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A5 be Subset of X5; :: thesis: for A6 being Subset of X6
for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A6 be Subset of X6; :: thesis: for A7 being Subset of X7
for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A7 be Subset of X7; :: thesis: for A8 being Subset of X8
for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A8 be Subset of X8; :: thesis: for A9 being Subset of X9
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let A9 be Subset of X9; :: thesis: for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]; :: thesis: ( x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] implies ( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 ) )
assume A1: x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] ; :: thesis: ( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 )
then A2: ( A1 <> {} & A2 <> {} & A3 <> {} & A4 <> {} & A5 <> {} & A6 <> {} & A7 <> {} & A8 <> {} & A9 <> {} ) by Th61;
reconsider y = x as Element of [:A1,A2,A3,A4,A5,A6,A7,A8,A9:] by A1;
( y `1 in A1 & y `2 in A2 & y `3 in A3 & y `4 in A4 & y `5 in A5 & y `6 in A6 & y `7 in A7 & y `8 in A8 & y `9 in A9 ) by A2;
hence ( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 & x `5 in A5 & x `6 in A6 & x `7 in A7 & x `8 in A8 & x `9 in A9 ) by Th84; :: thesis: verum