let s1, s2 be complex-valued ManySortedSet of NAT ; :: thesis: ( s1 . 0 = s . 0 & ( for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) ) & s2 . 0 = s . 0 & ( for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ) implies s1 = s2 )

assume that

A5: s1 . 0 = s . 0 and

A6: for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) and

A7: s2 . 0 = s . 0 and

A8: for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ; :: thesis: s1 = s2

defpred S_{1}[ Nat] means s1 . $1 = s2 . $1;

A9: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[ 0 ]
by A5, A7;

for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A10, A9);

hence s1 = s2 ; :: thesis: verum

assume that

A5: s1 . 0 = s . 0 and

A6: for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) and

A7: s2 . 0 = s . 0 and

A8: for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ; :: thesis: s1 = s2

defpred S

A9: for k being Nat st S

S

proof

A10:
S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume s1 . k = s2 . k ; :: thesis: S_{1}[k + 1]

hence s1 . (k + 1) = (s2 . k) + (s . (k + 1)) by A6

.= s2 . (k + 1) by A8 ;

:: thesis: verum

end;assume s1 . k = s2 . k ; :: thesis: S

hence s1 . (k + 1) = (s2 . k) + (s . (k + 1)) by A6

.= s2 . (k + 1) by A8 ;

:: thesis: verum

for n being Nat holds S

hence s1 = s2 ; :: thesis: verum