let x be real number ; :: thesis: ( 0 < x & x < 1 implies 0 < (1 - (x ^2 )) ^2 )
assume A1: ( 0 < x & x < 1 ) ; :: thesis: 0 < (1 - (x ^2 )) ^2
then x ^2 < x by SQUARE_1:75;
then x ^2 < 1 by A1, XXREAL_0:2;
then (x ^2 ) + (- (x ^2 )) < 1 + (- (x ^2 )) by XREAL_1:10;
hence 0 < (1 - (x ^2 )) ^2 by SQUARE_1:74; :: thesis: verum