let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n, REAL st M1 is_less_than M2 & M2 is_less_than M3 holds
M1 is_less_than M3
let M1, M2, M3 be Matrix of n, REAL ; :: thesis: ( M1 is_less_than M2 & M2 is_less_than M3 implies M1 is_less_than M3 )
A1: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
assume A2: ( M1 is_less_than M2 & M2 is_less_than M3 ) ; :: thesis: M1 is_less_than M3
for i, j being Nat st [i,j] in Indices M1 holds
M1 * i,j < M3 * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * i,j < M3 * i,j )
assume [i,j] in Indices M1 ; :: thesis: M1 * i,j < M3 * i,j
then ( M1 * i,j < M2 * i,j & M2 * i,j < M3 * i,j ) by A2, A1, Def5;
hence M1 * i,j < M3 * i,j by XXREAL_0:2; :: thesis: verum

end;

hence M1 is_less_than M3 by Def5; :: thesis: verum