let R be Ring; :: thesis: for n, i, j being Nat st i < j holds
for r being Element of R holds (1. (R,n)) + ((canMat (i,j,n,R)) * r) is upper_triangular
let n be Nat; :: thesis: for i, j being Nat st i < j holds
for r being Element of R holds (1. (R,n)) + ((canMat (i,j,n,R)) * r) is upper_triangular
let i, j be Nat; :: thesis: ( i < j implies for r being Element of R holds (1. (R,n)) + ((canMat (i,j,n,R)) * r) is upper_triangular )
assume A1: i < j ; :: thesis: for r being Element of R holds (1. (R,n)) + ((canMat (i,j,n,R)) * r) is upper_triangular
let r be Element of R; :: thesis: (1. (R,n)) + ((canMat (i,j,n,R)) * r) is upper_triangular
set M = (1. (R,n)) + ((canMat (i,j,n,R)) * r);
let i9, j9 be Nat; :: according to MATRIX_1:def 12 :: thesis: ( not [i9,j9] in Indices ((1. (R,n)) + ((canMat (i,j,n,R)) * r)) or i9 <= j9 or ((1. (R,n)) + ((canMat (i,j,n,R)) * r)) * (i9,j9) = 0. R )
assume A2: [i9,j9] in Indices ((1. (R,n)) + ((canMat (i,j,n,R)) * r)) ; :: thesis: ( i9 <= j9 or ((1. (R,n)) + ((canMat (i,j,n,R)) * r)) * (i9,j9) = 0. R )
assume A3: i9 > j9 ; :: thesis: ((1. (R,n)) + ((canMat (i,j,n,R)) * r)) * (i9,j9) = 0. R
A4: [i9,j9] in Indices (1. (R,n)) by A2, MATRIX_0:26;
A5: [i9,j9] in Indices ((canMat (i,j,n,R)) * r) by A2, MATRIX_0:26;
A6: (canMat (i,j,n,R)) * r is upper_triangular by A1, Th67;
thus ((1. (R,n)) + ((canMat (i,j,n,R)) * r)) * (i9,j9) = ((1. (R,n)) * (i9,j9)) + (((canMat (i,j,n,R)) * r) * (i9,j9)) by A4, MATRIX_1:def 8
.= ((1. (R,n)) * (i9,j9)) + (0. R) by A3, A5, A6
.= 0. R by A3, A4, MATRIX_1:def 3 ; :: thesis: verum