let a, b, c, d be ExtReal; :: thesis: ( a < b & c < d implies min (a,c) < min (b,d) )

assume that

A1: a < b and

A2: c < d ; :: thesis: min (a,c) < min (b,d)

min (a,c) <= c by Th17;

then A3: min (a,c) < d by A2, Th2;

min (a,c) <= a by Th17;

then min (a,c) < b by A1, Th2;

hence min (a,c) < min (b,d) by A3, Def8; :: thesis: verum

assume that

A1: a < b and

A2: c < d ; :: thesis: min (a,c) < min (b,d)

min (a,c) <= c by Th17;

then A3: min (a,c) < d by A2, Th2;

min (a,c) <= a by Th17;

then min (a,c) < b by A1, Th2;

hence min (a,c) < min (b,d) by A3, Def8; :: thesis: verum