let x be Real; :: thesis: ( 0 < x & x < 1 implies x ^2 < 1 )
assume that
A1: 0 < x and
A2: x < 1 ; :: thesis: x ^2 < 1
x ^2 < x by A1, A2, SQUARE_1:13;
hence x ^2 < 1 by A2, XXREAL_0:2; :: thesis: verum