let i, j be Element of NAT ; :: thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f & len (mid f,i,j) = 1 holds
i = j
let D be non empty set ; :: thesis: for f being FinSequence of D st i in dom f & j in dom f & len (mid f,i,j) = 1 holds
i = j
let f be FinSequence of D; :: thesis: ( i in dom f & j in dom f & len (mid f,i,j) = 1 implies i = j )
assume i in dom f ; :: thesis: ( not j in dom f or not len (mid f,i,j) = 1 or i = j )
then A1: ( 1 <= i & i <= len f ) by FINSEQ_3:27;
assume j in dom f ; :: thesis: ( not len (mid f,i,j) = 1 or i = j )
then A2: ( 1 <= j & j <= len f ) by FINSEQ_3:27;
assume A3: len (mid f,i,j) = 1 ; :: thesis: i = j

per cases ( i <= j or i >= j ) ;

suppose A4: i <= j ; :: thesis: i = j

then 0 + 1 = (j -' i) + 1 by A1, A2, A3, JORDAN4:20;
then 0 = j - i by A4, XREAL_1:235;
hence i = j ; :: thesis: verum

end;

suppose A5: i >= j ; :: thesis: i = j

then 0 + 1 = (i -' j) + 1 by A1, A2, A3, JORDAN4:21;
then 0 = i - j by A5, XREAL_1:235;
hence i = j ; :: thesis: verum

end;

end;