let G be Go-board; :: thesis: Int (cell (G,(len G),(width G))) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) }
cell (G,(len G),(width G)) = (v_strip (G,(len G))) /\ (h_strip (G,(width G))) by GOBOARD5:def 3;
then A1: Int (cell (G,(len G),(width G))) = (Int (v_strip (G,(len G)))) /\ (Int (h_strip (G,(width G)))) by TOPS_1:17;
A2: Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } by Th16;
A3: Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } by Th13;
thus Int (cell (G,(len G),(width G))) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } c= Int (cell (G,(len G),(width G)))
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Int (cell (G,(len G),(width G))) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } )
assume A4: x in Int (cell (G,(len G),(width G))) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) }
then x in Int (v_strip (G,(len G))) by A1, XBOOLE_0:def 4;
then consider r1, s1 being Real such that
A5: x = |[r1,s1]| and
A6: (G * ((len G),1)) `1 < r1 by A3;
x in Int (h_strip (G,(width G))) by A1, A4, XBOOLE_0:def 4;
then consider r2, s2 being Real such that
A7: x = |[r2,s2]| and
A8: (G * (1,(width G))) `2 < s2 by A2;
s1 = s2 by A5, A7, SPPOL_2:1;
hence x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } by A5, A6, A8; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } or x in Int (cell (G,(len G),(width G))) )
assume x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } ; :: thesis: x in Int (cell (G,(len G),(width G)))
then A9: ex r, s being Real st
( x = |[r,s]| & (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) ;
then A10: x in Int (h_strip (G,(width G))) by A2;
x in Int (v_strip (G,(len G))) by A3, A9;
hence x in Int (cell (G,(len G),(width G))) by A1, A10, XBOOLE_0:def 4; :: thesis: verum