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

assume that

A1: a <= b and

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

d <= max (b,d) by Th25;

then A3: c <= max (b,d) by A2, Th2;

b <= max (b,d) by Th25;

then a <= max (b,d) by A1, Th2;

hence max (a,c) <= max (b,d) by A3, Def9; :: thesis: verum

assume that

A1: a <= b and

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

d <= max (b,d) by Th25;

then A3: c <= max (b,d) by A2, Th2;

b <= max (b,d) by Th25;

then a <= max (b,d) by A1, Th2;

hence max (a,c) <= max (b,d) by A3, Def9; :: thesis: verum