let G be V9() X_equal-in-line Y_equal-in-column Matrix of (); :: thesis: cell (G,(len G),()) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) }
A1: cell (G,(len G),()) = (v_strip (G,(len G))) /\ (h_strip (G,())) by GOBOARD5:def 3;
A2: h_strip (G,()) = { |[r,s]| where r, s is Real : (G * (1,())) `2 <= s } by Th22;
A3: v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 <= r } by Th19;
thus cell (G,(len G),()) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) } c= cell (G,(len G),())
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in cell (G,(len G),()) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) } )
assume A4: x in cell (G,(len G),()) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) }
then x in v_strip (G,(len G)) by ;
then consider r1, s1 being Real such that
A5: x = |[r1,s1]| and
A6: (G * ((len G),1)) `1 <= r1 by A3;
x in h_strip (G,()) by ;
then consider r2, s2 being Real such that
A7: x = |[r2,s2]| and
A8: (G * (1,())) `2 <= s2 by A2;
s1 = s2 by ;
hence x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `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,())) `2 <= s ) } or x in cell (G,(len G),()) )
assume x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) } ; :: thesis: x in cell (G,(len G),())
then A9: ex r, s being Real st
( x = |[r,s]| & (G * ((len G),1)) `1 <= r & (G * (1,())) `2 <= s ) ;
then A10: x in h_strip (G,()) by A2;
x in v_strip (G,(len G)) by A3, A9;
hence x in cell (G,(len G),()) by ; :: thesis: verum