environ vocabulary PRE_TOPC, EUCLID, FINSEQ_1, GOBOARD1, RELAT_1, TOPREAL1, FINSEQ_4, BOOLE, MATRIX_1, FUNCT_1, ABSVALUE, ARYTM_1, MCART_1, TREES_1, INCSP_1, SEQM_3, ORDINAL2, TARSKI; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XCMPLX_0, XREAL_0, NAT_1, RELAT_1, FUNCT_1, FINSEQ_1, PRE_TOPC, ABSVALUE, FINSEQ_4, MATRIX_1, EUCLID, TOPREAL1, GOBOARD1; constructors SEQM_3, NAT_1, ABSVALUE, TOPREAL1, GOBOARD1, FINSEQ_4, INT_1, MEMBERED, REAL_1, XBOOLE_0; clusters RELSET_1, STRUCT_0, EUCLID, INT_1, XREAL_0, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve p,p1,p2,q for Point of TOP-REAL 2, f,g,g1,g2 for FinSequence of TOP-REAL 2, n,m,i,j,k for Nat, G for Go-board, x for set; theorem :: GOBOARD3:1 (for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j)) & f is one-to-one unfolded s.n.c. special implies ex g st g is_sequence_on G & g is one-to-one unfolded s.n.c. special & L~f = L~g & f/.1 = g/.1 & f/.len f = g/.len g & len f <= len g; theorem :: GOBOARD3:2 (for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j)) & f is_S-Seq implies ex g st g is_sequence_on G & g is_S-Seq & L~f = L~g & f/.1 = g/.1 & f/.len f = g/.len g & len f<=len g;