let z1, z2 be Tuple of n, BOOLEAN ; :: thesis: ( ( for i being Nat st i in Seg n holds
z1 /. i = IFEQ (i,1,TRUE,FALSE) ) & ( for i being Nat st i in Seg n holds
z2 /. i = IFEQ (i,1,TRUE,FALSE) ) implies z1 = z2 )
assume that
A6: for i being Nat st i in Seg n holds
z1 /. i = IFEQ (i,1,TRUE,FALSE) and
A7: for i being Nat st i in Seg n holds
z2 /. i = IFEQ (i,1,TRUE,FALSE) ; :: thesis: z1 = z2
A8: len z1 = n by FINSEQ_1:def 18;
A9: len z2 = n by FINSEQ_1:def 18;
A10: dom z1 = Seg n by A8, FINSEQ_1:def 3;

now

let j be Nat; :: thesis: ( j in dom z1 implies z1 . j = z2 . j )
assume A12: j in dom z1 ; :: thesis: z1 . j = z2 . j
then A13: j in dom z2 by A9, A10, FINSEQ_1:def 3;
thus z1 . j = z1 /. j by A12, PARTFUN1:def 8
.= IFEQ (j,1,TRUE,FALSE) by A6, A10, A12
.= z2 /. j by A7, A10, A12
.= z2 . j by A13, PARTFUN1:def 8 ; :: thesis: verum

end;

hence z1 = z2 by A8, A9, FINSEQ_2:10; :: thesis: verum