deffunc H₁( Ordinal) -> Subset of [:(Day $1),(Day $1):] = Triangle $1;
deffunc H₂( object , Function-yielding c=-monotone Sequence) -> object = [( { ((((union (rng $2)) . [xL,(R_ $1)]) +' ((union (rng $2)) . [(L_ $1),yL])) +' (-' ((union (rng $2)) . [xL,yL]))) where xL is Element of L_ (L_ $1), yL is Element of L_ (R_ $1) : ( xL in L_ (L_ $1) & yL in L_ (R_ $1) ) } \/ { ((((union (rng $2)) . [xR,(R_ $1)]) +' ((union (rng $2)) . [(L_ $1),yR])) +' (-' ((union (rng $2)) . [xR,yR]))) where xR is Element of R_ (L_ $1), yR is Element of R_ (R_ $1) : ( xR in R_ (L_ $1) & yR in R_ (R_ $1) ) } ),( { ((((union (rng $2)) . [xL,(R_ $1)]) +' ((union (rng $2)) . [(L_ $1),yR])) +' (-' ((union (rng $2)) . [xL,yR]))) where xL is Element of L_ (L_ $1), yR is Element of R_ (R_ $1) : ( xL in L_ (L_ $1) & yR in R_ (R_ $1) ) } \/ { ((((union (rng $2)) . [xR,(R_ $1)]) +' ((union (rng $2)) . [(L_ $1),yL])) +' (-' ((union (rng $2)) . [xR,yL]))) where xR is Element of R_ (L_ $1), yL is Element of L_ (R_ $1) : ( xR in R_ (L_ $1) & yL in L_ (R_ $1) ) } )];
let S be Function-yielding c=-monotone Sequence; :: thesis: ( ( for B being Ordinal st B in dom S holds
ex SB being ManySortedSet of Triangle B st
( S . B = SB & ( for x being object st x in Triangle B holds
SB . x = [( { ((((union (rng (S | B))) . [xL,(R_ x)]) +' ((union (rng (S | B))) . [(L_ x),yL])) +' (-' ((union (rng (S | B))) . [xL,yL]))) where xL is Element of L_ (L_ x), yL is Element of L_ (R_ x) : ( xL in L_ (L_ x) & yL in L_ (R_ x) ) } \/ { ((((union (rng (S | B))) . [xR,(R_ x)]) +' ((union (rng (S | B))) . [(L_ x),yR])) +' (-' ((union (rng (S | B))) . [xR,yR]))) where xR is Element of R_ (L_ x), yR is Element of R_ (R_ x) : ( xR in R_ (L_ x) & yR in R_ (R_ x) ) } ),( { ((((union (rng (S | B))) . [xL,(R_ x)]) +' ((union (rng (S | B))) . [(L_ x),yR])) +' (-' ((union (rng (S | B))) . [xL,yR]))) where xL is Element of L_ (L_ x), yR is Element of R_ (R_ x) : ( xL in L_ (L_ x) & yR in R_ (R_ x) ) } \/ { ((((union (rng (S | B))) . [xR,(R_ x)]) +' ((union (rng (S | B))) . [(L_ x),yL])) +' (-' ((union (rng (S | B))) . [xR,yL]))) where xR is Element of R_ (L_ x), yL is Element of L_ (R_ x) : ( xR in R_ (L_ x) & yL in L_ (R_ x) ) } )] ) ) ) implies for A being Ordinal st A in dom S holds
No_mult_op A = S . A )
assume A1: for B being Ordinal st B in dom S holds
ex SB being ManySortedSet of H₁(B) st
( S . B = SB & ( for x being object st x in H₁(B) holds
SB . x = H₂(x,S | B) ) ) ; :: thesis: for A being Ordinal st A in dom S holds
No_mult_op A = S . A
let A be Ordinal; :: thesis: ( A in dom S implies No_mult_op A = S . A )
assume A2: A in dom S ; :: thesis: No_mult_op A = S . A
consider S1 being Function-yielding c=-monotone Sequence such that
A3: ( dom S1 = succ A & No_mult_op A = S1 . A ) and
A4: for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of H₁(B) st
( S1 . B = SB & ( for x being object st x in H₁(B) holds
SB . x = H₂(x,S1 | B) ) ) by Def12;
A5: succ A c= dom S by A2, ORDINAL1:21;
A6: for B being Ordinal st B in succ A holds
ex SB being ManySortedSet of H₁(B) st
( S . B = SB & ( for x being object st x in H₁(B) holds
SB . x = H₂(x,S | B) ) ) by A1, A5;
A7: ( succ A c= dom S & succ A c= dom S1 ) by A2, ORDINAL1:21, A3;
A8: S | (succ A) = S1 | (succ A) from SURREALR:sch 2(A7, A6, A4);
A in succ A by ORDINAL1:8;
hence No_mult_op A = S . A by A3, A8, FUNCT_1:49; :: thesis: verum