let s be FinSequence of A; :: thesis: ( s is ascending implies s is weakly-ascending )

assume s is ascending ; :: thesis: s is weakly-ascending

then A1: for n, m being Nat st n in dom s & m in dom s & n < m holds

s /. n <~ s /. m ;

for n, m being Nat st n in dom s & m in dom s & n < m holds

s /. n <= s /. m

assume s is ascending ; :: thesis: s is weakly-ascending

then A1: for n, m being Nat st n in dom s & m in dom s & n < m holds

s /. n <~ s /. m ;

for n, m being Nat st n in dom s & m in dom s & n < m holds

s /. n <= s /. m

proof

hence
s is weakly-ascending
; :: thesis: verum
let n, m be Nat; :: thesis: ( n in dom s & m in dom s & n < m implies s /. n <= s /. m )

assume that

A2: ( n in dom s & m in dom s ) and

A3: n < m ; :: thesis: s /. n <= s /. m

s /. n <~ s /. m by A1, A2, A3;

hence s /. n <= s /. m ; :: thesis: verum

end;assume that

A2: ( n in dom s & m in dom s ) and

A3: n < m ; :: thesis: s /. n <= s /. m

s /. n <~ s /. m by A1, A2, A3;

hence s /. n <= s /. m ; :: thesis: verum