environ
vocabularies HIDDEN, NUMBERS, SUBSET_1, REAL_1, FUNCT_1, GOBOARD5, TOPREAL1, STRUCT_0, EUCLID, RELAT_1, XXREAL_0, FINSEQ_1, GOBOARD2, TREES_1, TOPS_1, TARSKI, XBOOLE_0, PRE_TOPC, CONNSP_3, RELAT_2, GOBOARD9, CONNSP_1, ARYTM_3, CARD_1, RLTOPSP1, PARTFUN1, MATRIX_1, ZFMISC_1, MCART_1, ARYTM_1, RCOMP_1, NAT_1, CONVEX1, CHORD;
notations HIDDEN, TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, TOPREAL1, ORDINAL1, NUMBERS, DOMAIN_1, XCMPLX_0, XREAL_0, STRUCT_0, REAL_1, NAT_1, NAT_D, PRE_TOPC, FUNCT_1, PARTFUN1, FINSEQ_1, MATRIX_0, TOPS_1, CONNSP_1, RLVECT_1, RLTOPSP1, EUCLID, GOBOARD2, GOBOARD5, GOBOARD9, CONNSP_3, GOBRD10, XXREAL_0;
definitions TARSKI;
theorems TARSKI, NAT_1, FINSEQ_1, SPPOL_1, GOBOARD7, GOBOARD8, GOBOARD9, PRE_TOPC, CONNSP_1, SUBSET_1, EUCLID, CONNSP_3, TOPREAL1, GOBOARD5, TOPREAL3, MATRIX_0, GOBRD10, GOBRD11, ZFMISC_1, XBOOLE_0, XBOOLE_1, FINSEQ_3, XREAL_1, XXREAL_0, PARTFUN1, XREAL_0, NAT_D, ORDINAL1;
schemes MATRIX_0;
registrations SUBSET_1, RELSET_1, XXREAL_0, NAT_1, STRUCT_0, PRE_TOPC, EUCLID, GOBOARD2, GOBOARD5, GOBRD11, JORDAN1, XREAL_0, ORDINAL1;
constructors HIDDEN, DOMAIN_1, REAL_1, TOPS_1, CONNSP_1, GOBOARD2, GOBOARD9, CONNSP_3, GOBRD10, GOBOARD1, NAT_D, RELSET_1, CONVEX1;
requirements HIDDEN, NUMERALS, REAL, SUBSET, BOOLE, ARITHM;
equalities CONNSP_3, STRUCT_0;
expansions TARSKI;
Lm1:
for f being non constant standard special_circular_sequence holds (L~ f) ` = the carrier of ((TOP-REAL 2) | ((L~ f) `))
Lm2:
the carrier of (TOP-REAL 2) = REAL 2
by EUCLID:22;
Lm3:
for f being non constant standard special_circular_sequence holds
( Cl (Down ((LeftComp f),((L~ f) `))) is connected & Down ((LeftComp f),((L~ f) `)) = LeftComp f & Down ((LeftComp f),((L~ f) `)) is a_component )
Lm4:
for f being non constant standard special_circular_sequence holds
( Cl (Down ((RightComp f),((L~ f) `))) is connected & Down ((RightComp f),((L~ f) `)) = RightComp f & Down ((RightComp f),((L~ f) `)) is a_component )
Lm5:
for f being non constant standard special_circular_sequence
for i1, j1, i2, j2 being Nat st i1 <= len (GoB f) & j1 <= width (GoB f) & i2 <= len (GoB f) & j2 <= width (GoB f) & ( i2 = i1 + 1 or i1 = i2 + 1 ) & j1 = j2 & Int (cell ((GoB f),i1,j1)) c= (LeftComp f) \/ (RightComp f) holds
Int (cell ((GoB f),i2,j2)) c= (LeftComp f) \/ (RightComp f)
Lm6:
for f being non constant standard special_circular_sequence
for i1, j1, i2, j2 being Nat st i1 <= len (GoB f) & j1 <= width (GoB f) & i2 <= len (GoB f) & j2 <= width (GoB f) & ( j2 = j1 + 1 or j1 = j2 + 1 ) & i1 = i2 & Int (cell ((GoB f),i1,j1)) c= (LeftComp f) \/ (RightComp f) holds
Int (cell ((GoB f),i2,j2)) c= (LeftComp f) \/ (RightComp f)
theorem Th6:
for
f being non
constant standard special_circular_sequence for
i1,
j1,
i2,
j2 being
Nat st
i1 <= len (GoB f) &
j1 <= width (GoB f) &
i2 <= len (GoB f) &
j2 <= width (GoB f) &
i1,
j1,
i2,
j2 are_adjacent holds
(
Int (cell ((GoB f),i1,j1)) c= (LeftComp f) \/ (RightComp f) iff
Int (cell ((GoB f),i2,j2)) c= (LeftComp f) \/ (RightComp f) )