let i 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 holds

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

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

A1: cell (G,i,(width G)) = (v_strip (G,i)) /\ (h_strip (G,(width G))) by GOBOARD5:def 3;

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

then A2: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by Th20;

A3: h_strip (G,(width G)) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 <= s } by Th22;

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

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

then A9: ex r, s being Real st

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

then A10: x in h_strip (G,(width G)) by A3;

x in v_strip (G,i) by A2, A9;

hence x in cell (G,i,(width G)) by A1, A10, XBOOLE_0:def 4; :: thesis: verum

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

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

A1: cell (G,i,(width G)) = (v_strip (G,i)) /\ (h_strip (G,(width G))) by GOBOARD5:def 3;

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

then A2: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by Th20;

A3: h_strip (G,(width G)) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 <= s } by Th22;

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

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

assume A4: x in cell (G,i,(width G)) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 <= s ) }

then x in v_strip (G,i) by A1, XBOOLE_0:def 4;

then consider r1, s1 being Real such that

A5: x = |[r1,s1]| and

A6: ( (G * (i,1)) `1 <= r1 & r1 <= (G * ((i + 1),1)) `1 ) by A2;

x in 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 A3;

s1 = s2 by A5, A7, 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,(width G))) `2 <= s ) } by A5, A6, A8; :: thesis: verum

end;assume A4: x in cell (G,i,(width G)) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 <= s ) }

then x in v_strip (G,i) by A1, XBOOLE_0:def 4;

then consider r1, s1 being Real such that

A5: x = |[r1,s1]| and

A6: ( (G * (i,1)) `1 <= r1 & r1 <= (G * ((i + 1),1)) `1 ) by A2;

x in 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 A3;

s1 = s2 by A5, A7, 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,(width G))) `2 <= s ) } by A5, A6, A8; :: 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,(width G))) `2 <= s ) } ; :: thesis: x in cell (G,i,(width G))

then A9: ex r, s being Real st

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

then A10: x in h_strip (G,(width G)) by A3;

x in v_strip (G,i) by A2, A9;

hence x in cell (G,i,(width G)) by A1, A10, XBOOLE_0:def 4; :: thesis: verum