let f be non constant standard special_circular_sequence; :: thesis: for i, j being Element of NAT st i > j & ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) holds
mid f,i,j is S-Sequence_in_R2
let i, j be Element of NAT ; :: thesis: ( i > j & ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) implies mid f,i,j is S-Sequence_in_R2 )
assume that
A1: i > j and
A2: ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) ; :: thesis: mid f,i,j is S-Sequence_in_R2
A3: Rev (mid f,j,i) = mid f,i,j by JORDAN4:30;

per cases ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) by A2;

suppose ( 1 < j & i <= len f ) ; :: thesis: mid f,i,j is S-Sequence_in_R2

then mid f,j,i is S-Sequence_in_R2 by A1, JORDAN4:52;
hence mid f,i,j is S-Sequence_in_R2 by A3, SPPOL_2:47; :: thesis: verum

end;

suppose A4: ( 1 <= j & i < len f ) ; :: thesis: mid f,i,j is S-Sequence_in_R2

then i + 1 <= len f by NAT_1:13;
then mid f,j,i is S-Sequence_in_R2 by A1, A4, JORDAN4:51;
hence mid f,i,j is S-Sequence_in_R2 by A3, SPPOL_2:47; :: thesis: verum

end;

end;