let i, j be Nat; :: thesis: for G being V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2) st 1 <= i & i < len G & 1 <= j & j < width G holds

cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }

let G be V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } )

assume that

A1: ( 1 <= i & i < len G ) and

A2: ( 1 <= j & j < width G ) ; :: thesis: cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }

A3: h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by A2, Th23;

A4: cell (G,i,j) = (v_strip (G,i)) /\ (h_strip (G,j)) by GOBOARD5:def 3;

A5: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by A1, Th20;

thus cell (G,i,j) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } c= cell (G,i,j)

assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } ; :: thesis: x in cell (G,i,j)

then A11: ex r, s being Real st

( x = |[r,s]| & (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) ;

then A12: x in h_strip (G,j) by A3;

x in v_strip (G,i) by A5, A11;

hence x in cell (G,i,j) by A4, A12, XBOOLE_0:def 4; :: thesis: verum

cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }

let G be V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } )

assume that

A1: ( 1 <= i & i < len G ) and

A2: ( 1 <= j & j < width G ) ; :: thesis: cell (G,i,j) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }

A3: h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by A2, Th23;

A4: cell (G,i,j) = (v_strip (G,i)) /\ (h_strip (G,j)) by GOBOARD5:def 3;

A5: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by A1, Th20;

thus cell (G,i,j) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } c= cell (G,i,j)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } or x in cell (G,i,j) )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in cell (G,i,j) or x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } )

assume A6: x in cell (G,i,j) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }

then x in v_strip (G,i) by A4, XBOOLE_0:def 4;

then consider r1, s1 being Real such that

A7: x = |[r1,s1]| and

A8: ( (G * (i,1)) `1 <= r1 & r1 <= (G * ((i + 1),1)) `1 ) by A5;

x in h_strip (G,j) by A4, A6, XBOOLE_0:def 4;

then consider r2, s2 being Real such that

A9: x = |[r2,s2]| and

A10: ( (G * (1,j)) `2 <= s2 & s2 <= (G * (1,(j + 1))) `2 ) by A3;

s1 = s2 by A7, A9, SPPOL_2:1;

hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by A7, A8, A10; :: thesis: verum

end;assume A6: x in cell (G,i,j) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }

then x in v_strip (G,i) by A4, XBOOLE_0:def 4;

then consider r1, s1 being Real such that

A7: x = |[r1,s1]| and

A8: ( (G * (i,1)) `1 <= r1 & r1 <= (G * ((i + 1),1)) `1 ) by A5;

x in h_strip (G,j) by A4, A6, XBOOLE_0:def 4;

then consider r2, s2 being Real such that

A9: x = |[r2,s2]| and

A10: ( (G * (1,j)) `2 <= s2 & s2 <= (G * (1,(j + 1))) `2 ) by A3;

s1 = s2 by A7, A9, SPPOL_2:1;

hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by A7, A8, A10; :: thesis: verum

assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } ; :: thesis: x in cell (G,i,j)

then A11: ex r, s being Real st

( x = |[r,s]| & (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) ;

then A12: x in h_strip (G,j) by A3;

x in v_strip (G,i) by A5, A11;

hence x in cell (G,i,j) by A4, A12, XBOOLE_0:def 4; :: thesis: verum